The underlying legal basis for Hezekiah's action is found in Numbers:
The LORD spoke to Moses, saying, “Speak to the people of Israel, saying, If any one of you or of your descendants is unclean through touching a dead body, or is on a long journey, he shall still keep the Passover to the LORD. In the second month on the fourteenth day at twilight they shall keep it... (Numbers 9:9-11 ESV)
The principle is the Passover may be observed in the second month if there is a "legal" reason why a person was unable to observe it in the first month.
The incident which prompted Moses to seek an answer from the LORD was a man who had touched a dead body:
And there were certain men who were unclean through touching a dead body, so that they could not keep the Passover on that day, and they came before Moses and Aaron on that day. And those men said to him, “We are unclean through touching a dead body. Why are we kept from bringing the LORD's offering at its appointed time among the people of Israel?” And Moses said to them, “Wait, that I may hear what the LORD will command concerning you.” (Numbers 9:6-8)
The answer Moses received added "being on a long journey" as a reason. This foreshadows a change to how the Passover would be observed when the land was occupied:
1 “Observe the month of Abib and keep the Passover to the LORD your God, for in the month of Abib the LORD your God brought you out of Egypt by night. 2 And you shall offer the Passover sacrifice to the LORD your God, from the flock or the herd, at the place that the LORD will choose, to make his name dwell there. 3 You shall eat no leavened bread with it. Seven days you shall eat it with unleavened bread, the bread of affliction—for you came out of the land of Egypt in haste—that all the days of your life you may remember the day when you came out of the land of Egypt. 4 No leaven shall be seen with you in all your territory for seven days, nor shall any of the flesh that you sacrifice on the evening of the first day remain all night until morning. 5 You may not offer the Passover sacrifice within any of your towns that the LORD your God is giving you, 6 but at the place that the LORD your God will choose, to make his name dwell in it, there you shall offer the Passover sacrifice, in the evening at sunset, at the time you came out of Egypt. 7 And you shall cook it and eat it at the place that the LORD your God will choose. And in the morning you shall turn and go to your tents. (Deuteronomy 16:1-7)
As Bernard M. Levensen explains:
The blend of Passover and the Festival of Unleavened Bread is the most remarkable section of this calendar. Passover was originally a separate observance, celebrated within the family or clan. Since it did not require a sacrifice at the sanctuary, it was not included among the three pilgrimage festivals…But Deuteronomy’s restriction of sacrifice to the single sanctuary prohibited Passover from being observed locally (v.5) and required that the observance be redirected to the central sanctuary (vv.2,6-7) 1
Hezekiah's primary dilemma is not from touching a dead body; rather from not having enough priests who were able to offer the Passover sacrifice. Thus he is unable to observe the Passover as instructed in Deuteronomy. However, the event in Numbers which occurs before there is a central sanctuary, does offer a legal precedent for observing the Passover in the second month.
Secondarily, "the people had not gathered themselves to Jerusalem." While this does not qualify as "being on a long journey," it does follow the general train of thought. Significantly, this was not part of the question Moses brought to the LORD; it was added by the LORD Himself. So it is not unreasonable for Hezekiah to recognize the primary intent is to ensure a person can observe the Passover, if there is an "extenuating circumstance" especially if there is something which is beyond an individual's ability to control. In this case not enough priests to observe the Passover as required in Deuteronomy.
Mercy not Sacrifice
With respect to the Passover and the Feast of Unleavened Bread which follows, I believe the statement Jesus makes "I desire mercy not sacrifice" (Matthew 9:13) points to His Passover sacrifice. In other words, the Scripture calls for people to be merciful because He will make the sacrifice. The context for the quote actually points to the Last Supper:
And as Jesus reclined at table in the house, behold, many tax collectors and sinners came and were reclining with Jesus and his disciples. And when the Pharisees saw this, they said to his disciples, “Why does your teacher eat with tax collectors and sinners?” (Matthew 9:10-11)
John provides the practical means by which disciples are to show mercy:
A new commandment I give to you, that you love one another: just as I have loved you, you also are to love one another. By this all people will know that you are my disciples, if you have love for one another.” (John 13:34-35)
The mercy Jesus desires is to love one another because of the sacrifice He is about to make on behalf of all people.
The annual calendar did not require people to observe the Passover in Jerusalem; it did require their presence for the next seven days. This would ensure everyone was present on the Day of First Fruits, the day of the Resurrection. This would also permit everyone to go and see if the tomb was empty and to hear, or experience what took place after the resurrection:
The tombs also were opened. And many bodies of the saints who had fallen asleep were raised, and coming out of the tombs after his resurrection they went into the holy city and appeared to many. (Matthew 27:52-53)
Observing the LORD's Passover in Jerusalem and His calendar required all Israel to be in Jerusalem when the Resurrection occurred.
Finally, the ability to observe the second Passover in the case of touching a dead body would allow those who buried Jesus to observe the Passover in the second month. This further shows He planned His death and ensured those who prepared His body in death would still be able to observe this important remembrance of the LORD freeing His people from bondage.
1. Bernard M. Levenson, The Jewish Study Bible, Oxford University Press, 2004, p. 401