I am developing a campaign were the party is tracking down a group of ancient powerful mages. These mages would have undoubtedly sought undeath to extend their existence and I didn't just want them to all become liches. I wanted to see if there are other alternatives that are likely (such as Vampire or Mummy Lord) that make sense.
You seem to have found all the contenders. The only undead in 5e which are intelligent, independent, and you can deliberately become are liches and mummy lords. To be a vampire requires another vampire, and you would then be subservient to the original vampire anyway. There are other kinds of powerful undead (such as wights or revenants) but none that you can deliberately and purposefully become.
Note that it's impossible for a single mage to become a vampire or free-willed vampire spawn, because death will end their spells, and the vampire they were trying to use to become a vampire will control them. However, with the help of another mage, it becomes very simple. Mage A casts Dominate Monster on a vampire, then forces the vampire to drain Mage B's blood and then give Mage B some of the vampire's own blood. If Mage A also wanted to become a vampire, Mage B could make Mage A one easily. (Although that requires a lot of trust between them, because Mage B has the chance of having complete control over Mage A, which most powerful evil mages would jump at.)
Lichdom is the usual method of immortality for powerful evil mages. For a bit of variety, some of them could have let themselves go, (as it were), and become demiliches. Likewise, some could have deliberately become demiliches as described in the sidebar on Acererak. A really clever mage could even have transformed themself into a dracolich, with a bit of work and some help from some of his friends. (True Polymorph plus the ritual to turn a dragon into a dracolich.)
However, it's your campaign, and you are free to do whatever you want. If you don't want your mages to be liches or mummy lords, they don't have to be. You can change the lore for an existing undead, or just create your own to suit your needs.
Just to pile on even more options on top of other answers and my earlier comment, you can get creative with existing things, like using the Clone spell repeatedly: Does Clone spell makes someone effectively immortal?
Or having someone cast Reincarnate on you every 30-60 years or so, to give you a new body.
You could live in the Astral plane, where you don't age (plus pesky adventurers can't get to you easily).
And let's not forget the simple approach of just being an elf. While this doesn't make you immortal, your lifespan is so long you're effectively immortal to all those other petty races (if that's the sort of superiority your villains are looking for).
If you are comfortable with planar travel, there are other options outside of being undead you can consider as well.
These mages may have gone to Elysium and secured water from the sacred river. They could also have visited Asgard and gotten themselves some golden apples. They may even have entered some of the lower plans and made pacts with demons, fiends, dieties, etc. With the addition of planar travel you can also simply invent things on other planes which offer extended life spans.
Looking outside of things which are "undead" opens a host of other options.
Have you considered other races? There's the Illithilich (Also called the Alhoon, or Mindflayer Lich), the Balenorn (Elf Lich, though not truly evil, they can be neutral), Psionic Lich, Bane Lich (Clerical, but still worth considering), Beholder Lich, Suel Lich, Lichfiend. There are more, I'm sure, but the point is that you could make the Lich theme more interesting by having some of the Magi be from other races.
There is also the many Kyuss templates you could apply to effectively create an undead mage and spawn of Kyuss.
There are many beings which live for incredible lifespans which don't require seeking undeath. Dragons are the first that come to mind, but there are others. The Spellweavers are effectively immortal, and are known to disguise themselves as humans to accomplish their unknowable aims. Devils could also pose as human mages, perhaps after replacing their summoner who failed to inscribe adequate protections, see: How can I make a deal with a devil.
Combinations of the above are also possible, for inspiration just look at the Age of Worms adventure path.
You could also have them cast true Polymorph on themselves turning into beings that either don't age or have longer lifespans. Such as dragons. Or you could have some research other ways of immortality. There are magical areas that slow time relative to everywhere else (dbz time capsule style), or perhaps they were able to turn themselves into ent like beings who take the feature of red woods that live for freakin ever. You could have them freeze themselves to be woken up in the future. Or perhaps they made a wish spell, or magic contract with a demon lord for immortality. All viable options.