Certified GKH Gardening Expert
The best ways to dry water saturated potting soil is of course protect the pot from additional rain or irrigation. Move it under cover or put up an umbrella.
If there is a saucer under the pot, dump out or spoon out excess water that drains through.
Exposure to sunlight and wind, even a fan, will help in the drying of the soil.
Monitor soil moisture content with your fingers or a trowel or better yet a soil moisture meter.