We used to get a maximum of about 6" standing water in the chicken pen where I live. We went the deep litter route.
We put down a full 1' of wood chips through the whole thing, raked in every leaf in our lawn, and threw down a bale of straw. Instant relief.
The wood chips came from local tree companies where you can get them for free. If you have the time/space, let them age for 6 months first to let any fungal blooms die down. We had 14 yards delivered to my property for free and dumped in our driveway. We move it about 150' manually. Takes about 4 people 2 hours but it's just one day and we buy food and ask friends to help and it pays off. No smell. No mess. No soggy ground. The wood chips soaked it all up like a sponge.
The payoff is great too. Not only do the chickens absolutely love it (digging through it, dust bathing in the fine particulate, etc) but every couple years we rake the woodchips off to one side and take the amazing new super-fertile topsoil out for a new garden bed. We could leave it there to keep raising the moisture tolerance of our soil and the grade of the chicken pen, but it's actually become a benefit to have all the water drain into the pen where it soaks into the deep litter. Our whole lawn is drier as a result and we have no swampy gross chicken mess.
Every couple years we get another load for free and move it into the chicken pen in large quantities. Works great. Of course, a drain works too, but for us there's nowhere to drain to. The whole area is fairly flat former swampland. And if you only have about 6" or less of standing water that stinks it's a good solution.