Re-casting Find Familiar isn't necessary unless the familiar was killed. If recast, the original familiar reappears.
The spell descriptions strongly indicate that re-casting Find Familiar re-calls the original familiar, but doing so isn't necessary unless the familiar was killed (or, at least, dropped to 0 HP).
Reincarnate creates a new body, but it's the same person inside
The description of the Reincarnate spell reads, in part:
[...] the spell forms a new adult body for [the dead creature] and then calls the soul to enter that body [...]
The magic fashions a new body for the creature to inhabit [...]
...which, to my reading, means that the reincarnated creature is the same person as the original, just with a different meat suit:
The reincarnated creature recalls its former life and experiences. It retains the capabilities it had in its original form, except [its race changes.]
The bond from Find Familiar is unaffected by the caster's temporary death
The Find Familiar spell has a duration of "Instantaneous". This tells us that we don't have to worry about any questions of the spell's duration wearing off or otherwise ending when the caster dies. The spell's description states, in part:
When the familiar drops to 0 hit points, it disappears, leaving behind no physical form. It reappears after you cast this spell again.
This tells us that the familiar itself is, at best, hard to kill. It also tells us that re-casting Find Familiar will re-call the original familiar.
The spell description adds:
Alternatively, you can dismiss it forever.
This tells us that there's a way the familiar/caster bond to be broken, but strongly indicates that the caster needs to do so intentionally.
In addition, the description specifies:
You can’t have more than one familiar at a time. If you cast this spell while you already have a familiar, you instead cause it to adopt a new form.
This reminds us that re-casting Find Familiar re-summons the existing familiar.
Nothing in the spell description specifies what happens to the familiar when the caster dies, which means that nothing special happens when the caster dies.
Finally, the first line states:
You gain the service of a familiar, [...]
This says that it's you gaining the familiar.
A reincarnated creature "retains the capabilities it had in its original form", and one of those capabilities was the ability to recall its familiar, either from the pocket dimension to which it can be temporarily dismissed or via re-casting Find Familiar.
I think the spell description is pretty clear in that the familiar you call with Find Familiar is the same each time (possibly excepting the case in which the caster has chosen to dismiss the familiar forever).
So, my reading is that the spells are pretty clear:
The reincarnated caster can recall their familiar (possibly requiring an action to temporarily dismiss it to its pocket dimension first) upon being reincarnated. Further, if they re-cast Find Familiar for any reason, without having chosen to dismiss their existing familiar forever, they get the same familiar back.