If it is the first time, the body will take some time to make specific anti-antibodies, the antivenom has plenty of time to bind to the poison.

On a future second time (specially if the first was recent) you could get a race between the kinetics of your anti-antibodies binding the antivenom and the antibodies binding the poison. I would expect it would still work, but with decreased effect.

Finally, and this is the way that it always works, the poison + antibodies (yours or external) end up making a bigger clump that is consumed by cells of the immune system (macrophages), so ultimately you always get an immune response (but that does not necessarily mean an allergic reaction)

Some patients develop a sensitivity to anti-venom. However, there is a common myth that you cannot have anti-venom more than once. It is totally false. If you do develop a sensitivity and have a reaction from a subsequent administration of anti-venom, that reaction is very easily managed in a hospital setting.

Addressing your comment below, anti-venom does not provide much if any lasting benefit. Snakebite vaccinations have not shown effective in humans or pets. First, the venom antibodies are relatively short-lived. Second, vaccinations work well when your immune system gets to have a fair foot race with an incubating infection. Snake bites don’t work that way. You can be delivered a lethal dose in half a second. No amount of vaccination can prep your body to catch up with a sudden envenomation.