Already usually goes in the middle of a sentence: I have already booked your ticket. It is possible, however, to end a sentence with already, like you did in your sentence.
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I don't have a problem with the placement of the adverb.
Why do you think this is wrong in the first place? You already got an answer to the one with "I booked" and were told that "I have booked" is the usual way to say it.