The correct way to say it is:
"Llámame cuando salga del hospital."
The "technical" reason, going into grammar, would be that you're talking about an action that's not necessarily certain, and if it happens, it will be completed in the future, so you need to use the tense called "subjuntivo" in Spanish, and in this case you use the "subjuntivo presente", which can be used for present or future actions (even though the name says "present").
But you're doing the right thing by asking, because as it happens with languages, sometimes the rules overlap, or don't apply to all cases, so going by the grammar rules sometimes doesn't give you an unambiguous answer. It's better to ask how most people say it, and then try to learn with which expressions certain tenses are used.