If the idea of reincarnation is biblically supported, then how can that be reconciled with the idea of hell?
Even if souls are immortal, then they can possibly still be trapped (Psalm 57:6, 124:7), maybe broken up into pieces (of coe, presumably, Matthew 21:44), bought and sold (Revelation 18:13), made impure (Ezekiel 4:14), and/or cut off from other souls (Genesis 17:14). Each of these implies that the soul can recover and/or be recovered from its former status. Even broken things can be put back together again, or at least enough to be similar to its former self.
As for hell, in Matthew 10:28, the word that has been translated "destroy" is actually likely a mistranslation. A similar word, but a different version, is used in Matthew 10:6, referring to sheep of the house of Israel. Why would Christ send his disciples to sheep that have been destroyed? The word likely has the connotation of lost or even broken and scattered, or any of the aforementioned statuses, from which a soul can be recovered.
A very similar version of this Ancient Greek word, APOLESAS, is in Matthew 10:39. There, it is said that the PsYChEN or mind (related to the english word "psyche") can be found again after it has been APOLESAS. With GEENNE or Gehenna or hell, those unfortunate statuses would likely come upon souls in it, but they may not be able to recover themselves, and if others don't want to recover them, then they may remain in those unfortunate statuses and/or worsen. In most of these cases, reincarnation can apply. But in Gehenna, amounts of coe in souls, may be reincarnated into bad situations each cycle.
This calls to mind ideas like: "gates of heaven closed and protected against invaders." And "separating light from darkness." But also Luke 16:9, possibly about making friends who are more righteous or better off reincarnationally than one's self, in order to have friends who may (help) recover the self from unfortunate statuses, and/or help one's self reincarnate (or move during life) to better distributions, such as eternal dwellings, even if the self was currently not in Gehenna.
And even if two evil people decide to help each other stay out of hell and reincarnate to better lives, any other entities and networks of entities may notice that one helped the other, and both may be punished subsequently. And when I refer to evil, then remember that even very very righteous people may have sinned once. It's best to firstly and ultimately rely on Christ, who is perfect, to help recover, and also whoever He sends, those working for Him.