The OP asks specifically about extra-biblical resources available for exploring the meaning of rare words in the Bible. The lexical situation is very different when considering Classical Hebrew and Koine Greek, so I will consider those separately. First, a few general remarks about Biblical word studies:
The best resource for English readers to understand the words of the Bible is the English translations. The OP mentions concern about settling for "what translators felt it to mean". It is admirable to aspire to more in-depth analysis, and the investigation itself can be edifying study, but the translators of modern texts have at their disposal many years of education and experience with the languages as well as the best available lexical resources (see below). Any conclusions from one's own word-study that contradict the consensus of translators should be be met with a healthy skepticism.
The next best resource for English readers (and, indeed, for all but the most skilled Hebrew and Greek scholars) to determine word meaning is the lexicons. It is the job of the lexicographer to peruse the available literature and summarize their findings. Because most lexicons of ancient Greek and Hebrew to some extent also acts as concordances (i.e. they cite examples from period literature), these are included below. We also have a Meta post with a guide to some of the online resources available in this area. As I compose this answer, I see we have another that reviews the available Greek lexicons.
I think you may be overestimating the available extra-biblical corpus of ancient Hebrew. To my understanding, we have:
This category includes texts from diverse geographical sites and spanning a long period of time; however, the total amount of text is limited. As of 2009, the text of the Inscriptions amounted to about 2% of the size of the Hebrew Bible.1
Ben Sira was 2nd C. BCE Jewish scribe best known for the Wisdom of Sirach (=Ecclesiasticus). That book was only known in its Greek version until manuscripts (dating from the 11th-12th C.) were found in the Cairo geniza. Portions have also been found among the DSS. Together, the available text of Hebrew Ben Sira amounts to about 3% of the size of the Hebrew Bible.
Dead Sea Scrolls (DSS)
Our tag wiki provides a helpful introduction. Although, as pointed out there, about 80% of the manuscripts included in this group are non-Biblical, these are, for the most part, considerably shorter. Compiled, they amount to about 20% of the text of the Hebrew Bible.
As you can sea, the Hebrew Bible itself remains by far most plentiful source of Classical Hebrew text. The lexical resources in the Meta post provide information about this usage. To my knowledge, the only lexicon that also includes the extra-biblical sources (up to ~200 CE) is the Dictionary of Classical Hebrew, ed. David Clines. From the introduction:
Unlike all previous dictionaries of ancient Hebrew, this work does not restrict itself to, or privilege in any way, those ancient Hebrew texts found in the Hebrew Bible. Rather, it views Hebrew simply as a language like any other ancient language, for which it is necessary to examine the evidence of all the extant texts.
This formidable 8-volume work not only considers extra-biblical sources, it also comprehensively catalogues them in a syntagmatic analysis (i.e. listing all the subjects and objects that are attested for every verb, and, for nouns, all the verbs and all the other nouns with which they are connected). Unfortunately, to my knowledge this has no freely available version unless you have access through a library.
The situation with Greek is much different, and it's certainly true, as the OP suggests, that most hapax legomena within the Greek NT have attestation elsewhere in the corpus of Greek literature (both Ancient and Koine). In general, these sources have been long known and well catalogued. There are several lexicons that account for these data, and even those such as BDAG that focus on the NT frequently consider extra-biblical sources.
As an example, consider ἅλωσις in 2 Peter 2:12. Neither this term nor its cognate verb ἁλίσκομαι occur elsewhere in the NT. However, you can see that both are well attested in other literature. In the online Liddell-Scott-Jones Greek-English Lexicon (LSJ), many examples are given with active links to the Greek text. The LSJ entries provide representative examples but are not necessarily comprehensive.
If you would like to search the Greek literature yourself, the Thesaurus Linguae Graecae might be the best way. You need to register for a free account which will give you access to the "abridged" version. This is quite powerful. For instance, searching for the noun ἅλωσις ("lemma") gives you 254 results, each linked to the full text (follow the →). In order to make much out of this beyond a general sense of the distribution of usage, knowledge Greek is required.
Arguably more relevant to NT usage would be a survey of uses in the Septuagint. LSJ includes examples of LXX usages, and most of the "theological dictionaries" such as TLOT mentioned in another answer — also Kittel's TDNT and the more recent NIDNTTE — will include a survey of uses in Jewish literature, including the LXX.
1. All numerical data about the relative quantity of available Hebrew text are taken from the introduction the ש–ת volume of DCH, where the authors aggregate all of the words from every source for all volumes of the Dictionary.