I always have problem using the word already, or simply have trouble with tenses; in the first place, is the sentence in the question correct? secondly, when should we use already, can we use it in present tense?
Your sentence is correct as it stands. "Already" can be used with the present tense:
I am already in London.
I already play tennis, so I'd like to learn squash.
You could consider using the present perfect in your example:
You have to choose an hour that hasn't already been taken by someone else.