All "after July" tells me is that somebody plans to do something at some time after the month of July. If sunnyweather truly plans to look for work in August, then it would make more sense to say "In August".
I would rather hear "after July" than the longer, clumsier "when July finishes", which seemed to be sunnyweather's main concern in this question.
I'd really have to see the Polish student's sentence to comment on how normal "after July" seemed in that sentence.
In real life, I occasionally hear native speakers use phrases like "I'll do it sometime after July" when they talk about something they plan to do after the month of July has passed. It can be useful if the speaker doesn't know whether he will do it in August, September, November, or December.
Thank you. "After July" seems okay to me. Maybe this writer doesn't know exactly when he or she will begin looking for a new job. "After July" is vague enough to refer to any date that occurs after the end of July.
If you dont like the August option for some reason I would still look to rephrase the OP sentence. In answer to your question: "Does it look right?" my answer is "No". It is hard to say why, exactly, but it sounsd weird to me.
It works, but only if 'July' is understood by both speakers to have some significance.
"After the summer I'm going to look for a new job" sounds perfectly fine, for example. I would kind of expect you to say 'after my holidays' or 'after my exams' or some logical end of something that will free you up to go job-hunting. 'July' on its own doesn't quite do that.