Bomb it all!! Lol, just kidding :)

Don't kill off your potato bin just yet- the rain will slow down soon enough. Make sure anything you have standing water in is dumped off- or use mosquito dunks in things like pools or birdbaths. You can't really do anything about standing water in your yard, but that will drain down too.

If you are cool with chemicals- use a can or two of yard fogger in the area over the course of a week or two to help discourage the little suckers. You don't have to use an entire can each time- but fog the area good each time. Don't use spray bug killer that could get into your tater bin!

Or use a bug zapper by the bin. There's lots of options outside the plug into an outlet and fry the neighborhood zappers now. Sticky traps sometimes work with skeeters too.

If you can encourage skeeter eaters, do so. Bats and dragonflies like to eat skeeters.

Spend a little bit less time around your tater bin- and let nature take it's course.

Thanks for the tips! I have recently found a source of free black dirt that I can mix in with my clippings... Hopefully that will help somewhat to prevent the clippings from turning into a soggy matted mess. I wondered about chemicals but wasn't sure if there were any safe to use directly in the bin. Sticky traps would seem like a fairly cheap solution to give a try.

The skeeters are breeding in standing water, but the adults are holding around your potato bin. If you don't have standing water in the potato bin they aren't breeding there, but they are resting and ambushing from there, which is what your problem is and it is only the females at any rate. Problem is likely worst as light wanes in the evening; so work around this place more in the early afternoon or morning for fewer active mosquitoes. Different species come out to feed at different light intensities unless disturbed. The worst around here show up at dusk, the little striped vampires! Or they can be active in proper shade or if tall grass, etc., gets disturbed. There are a number of mechanical and chemical repellants that work pretty well, provided you remember to use them.