That's a good question. As a senior UX Designer I got to work both full-time on startup projects, got hired freelance to help startups review their UX and launch for milestone improvements.
There is a slightly new model that I would recommend, and have been successfully implementing with my startup clients.
The model is to bring a UX Designer on board on an ongoing basis, and even if it is with very minimal contribution at first.
I personally believe that great teams create great companies, which is why I would recommend that you think of that UX designer as part of your team. It is slightly different than you thinking of them and hiring them as an outsider working on a project, because once that project or phase is done, you and your developer will struggle with future iterations.
It would help for you to have a conversation with your potential UX designers. You would focus the conversation on one very small and specific area of your startup (ex: a section in the sign up page). It will be an opportunity for you to see how they think about UX problems. It would also provide you both with a sense of how it would be like to work together.
If it feels like a good fit, then give them a small UX design assignment that doesn't take more than a couple of hours for them to show a difference. If the result shows signs of a good direction, hire that designer to join you for as little as one hour a day. Then, you can increase that based on actual value and when you have a growing need for more of their contribution.
I would be happy to talk more about your startup, offer advice on the type of UX designer that could be a good fit and what simple next steps you could take that would make a difference.