I'm new to the whole tumble dryer business, and I recognize that if the clothes aren't folded immediately they will wrinkle completely. My question is, what is the most effective way to get rid of the wrinkles again. I've seen people suggesting just running them a bit more, throwing in a wet towel, etc. but I'm wondering whether the only true solution is to soak the clothes and run the drying process again, as energy inefficient as it seems.
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Steam is the best agent for getting wrinkles out of clothes, but dry heat comes in a distant second. Many newer dryers have a dedicated "touch-up" setting, which runs at medium-to-high heat for about ten minutes, precisely because it's very common to leave clothes in the dryer overnight. That touch-up will be a dry heat, of course, so it's not quite as effective as steam, but you'll be amazed at the difference it makes.
Adding a wet (wrung out or from the spin cycle - NOT sopping wet!) towel will improve your results significantly, but you'll need to 1) run the dryer for a bit longer to allow the wet towel to start steaming, and 2) steer a careful course between letting your other clothes get wet, and letting the whole batch get dry.
I would pop in the towel and start the dryer, then come back in 10-15 minutes and start pulling clothes out a few at a time - letting the dryer run in between while I folded or hung them up. Leave the towel for last; if you have another load waiting to dry, let it finish with that load.