I'm not sure, but I'd say that between 'full' and 'time' goes a hyphen. full - time job
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hirashinit's time to start looking for a full time job after I graduate.That doesn't really make sense It seems to say that it is time (i.e., now) to start looking for a job, and also that the time you will be looking for a job is after you graduate. The 'for' solves the problem.
moginnNow that I'm 21, it's time to start looking for a full time job by the time I graduate.The problem here is similar to the one in hirashin's sentence.
moginnFull-time job (with hyphen)That's correct. I missed it in my post.