I'm from USA. I know I can visit Thailand for 30 days without a tourist visa.
I accidentally booked a trip for 31 days - will they check my itinerary and deny access to the country because of this, without a visa?
It's unlikely they will deny entry, there are multiple systems in place to extend/pay for overstaying. For example, you can pay 1,900 baht while in Thailand to extend the 30 day duration. Also, if you overstay and are caught there is a 500 baht fine per day. So I'd say that when you land, if they notice, they will ask you to pay one of the above fees. Just explain the mistake and that it's only one day when you land.
Alternatively, you could do a short trip to Cambodia or Laos (great places to see!) then get another visa by crossing back into Thailand. It usually takes about a day to get to a border crossing from most Thai cities. about 1 year ago