Neither of these really qualifies as a contest, but these are places to submit poems where the deadlines are rolling, a bit like the weekly contest here.
Yellow Chair Review - Rock the Chair http://www.yellowchairreview.com/#!rock-the-chair-submission-guidelines/...
- no simultaneous submissions
- winner is published and receives $5
Poets Respond (Rattle) http://www.rattle.com/respond/
- poem about an event from the week
- $50 and publication to chosen poem