My answer might not be in line with the question if you only mean technical methods. However there is the obvious notion of prayer and reliance upon the Holy Spirit to open our eyes to what He has written to us. After all if the Spirit is the true author and men only the medium, we have more advantage in understanding the words by the Spirit than we would if we could talk to the original author. I would put this upon the same level as understanding the literal language itself. What I mean is:
Without a spirit of payer in our meditation of the living word, we are like fools reading a book from another language only guessing at what each syllable means.
-(1) Reading the whole Bible in a spirit of meditative prayer and submitting to what God manifests to us, through the Spirit, is the most effective means of understanding it.
Sample bible verses that make this a fundament principle of exegesis are:
Open my eyes that I may see wonderful things in your law. (NIV Psalm 119:18)
We are not like Moses, who would put a veil over his face to prevent the Israelites from seeing the end of what was passing away. But their minds were made dull, for to this day the same veil remains when the old covenant is read. It has not been removed, because only in Christ is it taken away. Even to this day when Moses is read, a veil covers their hearts. 16 But whenever anyone turns to the Lord, the veil is taken away. Now the Lord is the Spirit, and where the Spirit of the Lord is, there is freedom. And we all, who with unveiled faces contemplate the Lord’s glory, are being transformed into his image with ever-increasing glory, which comes from the Lord, who is the Spirit. (NIV 2 Corinthians 3:13-18)
I keep asking that the God of our Lord Jesus Christ, the glorious Father, may give you the Spirit of wisdom and revelation, so that you may know him better. I pray that the eyes of your heart may be enlightened in order that you may know the hope to which he has called you, the riches of his glorious inheritance in his holy people, and his incomparably great power for us who believe. (NIV Ephesians 1:17-19)
“This is the covenant I will make with the people of Israel after that time,” declares the Lord “I will put my law in their minds and write it on their hearts.
I will be their God, and they will be my people. No longer will they teach their neighbor, or say to one another, ‘Know the Lord,’ because they will all know me, from the least of them to the greatest,” declares the Lord. (Jeremiah 31:33-34)
After meditative prayer with expectation that God will manifest the meaning by His Spirit, I would put (not in strict order):
-(2) Understanding the language.
I do not mean a degree in the language. I simply mean recognizing each of the words in the text. The Bible uses a fairly extensive vocabulary and each word has special meaning for the Holy Spirit to have chosen it above others.
-(3) Knowing the ‘whole Bible’ to form a wide context and recognizing the gradual unfolding of the redemption plan in it.
The Messiah is the central point of which all other words stem. He is like the head of the body of scripture and when we twist any word away from the head we have dosconnected it from its organic place.
Do not let anyone who delights in false humility and the worship of angels disqualify you. Such a person also goes into great detail about what they have seen; they are puffed up with idle notions by their unspiritual mind. They have lost connection with the head, from whom the whole body, supported and held together by its ligaments and sinews, grows as God causes it to grow. (Coloassians 2:18-19)
-(4) Understanding what men of God from previous generations have thought.
-(5) Reaseraching related secular history surrounding any historical biblical texts, or making the symbols used in them more plain.
This ‘related history’ is usually very conveniently found under (4). However, I would caution that balance is needed here. Possibly 80% - 90% of scripture does not need any special historical background to fully follow the logic of the passages.
-(6) Understanding the language as it was 'originally written' in Hebrew or Greek.
I would put this much higher if the original texts were not already translated into our own language by learned people.
-(7) Using anything else that helps
Anything that helps, regardless of its origin, whether discoveries in science, archeology, art, even philosophy, your grandma’s jokes, anything in all the earth ---> if it helps, can be used.
Why we can use anything:
Although (7) amounts to very little help in most instances, they are still valid. Even appealing to bare ‘nature’ is used as valid exegesis by the Apostle Paul. Therefore, all those things I listed that seem not directly derived from within the Bible, can be covered under our God given natural reason, including everything man made that is derived from what is studied in nature. Basically, I am keeping ‘all doors open’, with little expectation from the result. I doubt any advancement in knowledge over the next one thousand years will have any impact on 99% of biblical interpretation. Also, most related history can already be gleaned from old commentaries. Therefore, whatever might appear "eisegetical" to some may just be a weaker form of what is "exegetical" ---> if it is reasonable. If it is 'persuasive' and has 'truth' and can be argued to relate to any given biblical text, then it can be associated with ‘reason’ to which the scripture pertains and of which comes from God. Of course, when anything claimed as reasonable, contradicts that which is clearly intened by the human author as derived from strictly ‘exegetical’ methods, such claims looses their persuasiveness immediately, upon the same reason and light of nature from which all they were pretended. In summary, 'any reasonable thing' can be used and is exegetical, for from nature comes natural reason, science, etc. and Paul argues this way from nature, here, for example:
18 The wrath of God is being revealed from heaven against all the godlessness and wickedness of people, who suppress the truth by their wickedness, 19 since what may be known about God is plain to them, because God has made it plain to them. 20 For since the creation of the world God’s invisible qualities—his eternal power and divine nature—have been clearly seen, being understood from what has been made, so that people are without excuse. (NIV Romans 1:18-20)
Back to the main point though, studying the Bible is primarily not an intellectual exercise but a 'spiritual encounter' with the voice of God. If we are not willing, hearing and submitting to His voice, our study will further confuse our minds. God's word never comes back void; it will harden or soften the heart. We will do what they did if we resist what is plain:
For although they knew God, they neither glorified him as God nor gave thanks to him, but their thinking became futile and their foolish hearts were darkened. Although they claimed to be wise, they became fools and exchanged the glory of the immortal God for images made to look like a mortal human being and birds and animals and reptiles.
Therefore God gave them over in the sinful desires of their hearts to sexual impurity for the degrading of their bodies with one another. They exchanged the truth about God for a lie, and worshiped and served created things rather than the Creator—who is forever praised. Amen. (NIV Romans 1:21-24)
A defense to this view
The worldly ‘objection’ that people might have to the 'self evincing' 'perfect word of God' by the 'illumination of the Holy Spirit', is that it exposes fools into claiming God has ‘taught them something’ special, like a private interpretation. For example, the cults, etc. However this objection is using incorrect exegesis in the meaning of 'private interpretation'. This objection has no relevance because fools will do that anyway, whether claiming 'private revelation' or 'private exegeses'. The fact is, since the days of Christ, simple bible believers have shared a common faith and it seems hardly fair to say such unanimous beliefs, derived from the overwhelming 'persuasiveness' of God’s word, under the 'power of the Holy Spirit' is ‘private’. On the contrary, it is private only if it is 'man made'. Unless the 'same Spirit' reveals 'to us' the 'same truths', that the Spirit who originally spoke through the prophet made known to the prophet (which therefore was not the prophets private opinion) can we say we do NOT have a private interpretation. Every other form of exegesis is a 'private interpretation', because it comes from the will of man and the understanding of his flesh, which are by definition not from God.
Above all, you must understand that no prophecy of Scripture came about by the prophet’s own interpretation of things. For prophecy never had its origin in the human will, but prophets, though human, spoke from God as they were carried along by the Holy Spirit. (2 Peter 1:20-21)
By the same Spirit who carried the prophet along, also carries us along into an understanding of the text. We are by this means to collect our manna direct from Him in a humble broken spirit, though we do have to 'get up' and 'find it' and 'prepare it' and that is much more difficult to us then it sounds:
If any of you lacks wisdom, you should ask God, who gives generously to all without finding fault, and it will be given to you. (NIV James 1:5)