The issue is most companies do a background check where everything listed on your resume is verified, so the dates of employment will be listed and overlapping dates would be a red flag
In my experience, most companies I've worked at are pretty slim with the background checks (in fact, having been given out as a primary reference on many of my friends/coworker's resumes), I get almost no contacts/calls following up and checking that reference.
In theory, this is something most companies would do, but anecdotally, it feels like nobody ever has the time.
You could just pick the one you want to present to the new prospective employer. This seems like a non-issue. It's like complaining about having a back pains, because you have to carry your wallet flush with cash.
