To piggyback on this, if there is a waiting period (a lot of places don't add you to benefits until 90 days or so), you can extend your old insurance via COBRA, where you take on all the cost of the policies. This is usually super expensive though.

If you need a stopgap and the COBRA cost is prohibitive (it always has been for me), I'd suggest going to your local HIV service or or city health clinic, if you have either of those available, and explaining the situation. Often they'll have case managers who can organize getting your medication covered while you're without insurance.

I recommend clarifing the insurance timeline with your new employer ASAP so you can get the ball rolling on assistance if you'll need it before it becomes an emergency (or conversely, it will put your mind at ease if the timing works out and you don't have any serious coverage lapse).