I've developed a close relationship with my past graduate teaching assistant, and I'm very interested in her: is it appropriate to pursue this?
Since you are not working under her supervision right now, then assuming you are both adults, you are in the clear ethically, legally, and university policy-wise.
The only real caveats are that if you end up involved in a romantic relationship with her, she will not be able to write you a letter of recommendation, and will not be able to assume a supervisory role over you in the future. However, academia has plenty of dating and married couples working in the same department and sometimes on the same project. There are well-established norms for dealing with such situations, and no one thinks this is anything to get worked up about.
Good luck!
Edit: as I somewhat feared would happen, people are volunteering unsolicited advice about whether it’s a smart idea to enter a relationship with someone you may end up sharing a workplace with in some hypothetical future, based on further hypotheticals about what would happen if you end up splitting up or whatnot. Since I respect you as a fully autonomous adult capable of making your own decisions, I have restricted my answer to things you actually asked about and that are in the scope of the sort of advice this site is designed to offer. I would advise others to do the same. There’s nothing inappropriate about a romantic relationship in your situation. It may be a good idea or a bad one, wise or unwise based on numerous factors as with any question about whether two people are a good match for each other. But that’s not what you asked about, and has no connection to academia.