I never did cover letters. I have never been out of work for more than a month.
I just tailored the resume to the job I was applying for. HR people do not like to read, managers like it even less. I was always taught that anything more than one page will go to the bottom of the pile if you are lucky. I don't know if that is true or not, just what I was told.
Unless this was advertised as a temporary position, do not make it look like you are after a temp job till you can go back to your "real" job. If it is a permanent position that could turn into a full time position. Treat it like you really want the job and plan to stay forever.