I have a habit of over-thinking stuff that leads me to worry about non-existent things like 90% of the time. I do the same thing. I create problems in my head, based on simply having received an email and I stress out over it for ages for no reason at all.
I think you should apply, and then if/when you get the job and you would have to start before this current job is over with, just be honest and say that you applied for a job because you knew this was ending and you needed something lined up, but you didn't know at the time that it would start before June 30th/that they wanted to extend your temping, and that sorry, but you're going to have to quit.
Which is easy for me to sit and suggest, but perhaps not so much to actually do. But, I really hope you won't hold a potentially awkward scene in which you will have to actually quit your temp job from applying to a job you really want.
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I think you should apply, and then if/when you get the job and you would have to start before this current job is over with, just be honest and say that you applied for a job because you knew this was ending and you needed something lined up, but you didn't know at the time that it would start before June 30th/that they wanted to extend your temping, and that sorry, but you're going to have to quit.
Which is easy for me to sit and suggest, but perhaps not so much to actually do. But, I really hope you won't hold a potentially awkward scene in which you will have to actually quit your temp job from applying to a job you really want.