**In Tasha's Cauldron of Everything there are Sidekick rules**

https://www.dndbeyond.com/sources/tcoe/dungeon-masters-tools#Sidekicks

This is probably your best bet for advancing a familiar. Talk to your DM, though.

Beyond that, there are not many options in 5e.

However, this has been done in previous editions.  So there is precedent out there.

If you were to go this route, you would need to check with your DM, of course.  So I mention previous editions because it is a great source for ideas on how to do this.  In the past this was done through:

- Spells
- Feats (Improved Familiar in 3.5 most notably)
- Rituals (Before rituals were spells)
- Magic Items 
- Prestige Classes 
- At a higher level you could often get a more powerful familiar (imp, pseudodragon, etc.)

Improved familiar stands out as probably the likeliest candidate for updating to 5e.

I will research my old docs to see if I can source these methods, but it might take some time.  If / When I find them, I'll post them here.

Here is one of my favorite articles on improving familiars from [Dragon #181][1] pg 22.  While it is for 2nd Ed, it is easily translatable to 5th, and it seems like a very reasonable place to start when building familiar improvements.

There are some [neat familiar feats here][2] Dragon # 374 pg 32 - but they are from 4th Ed and would take some more work to get them into shape.


  [1]: https://annarchive.com/files/Drmg181.pdf
  [2]: http://zebo.org/bruce/Dragon374.pdf