I would agree that if it is a new behavior and nothing has changed in his life otherwise, a vet check is definitely in order. Sometimes a refusal to eliminate in front of the owner but then going when no one is around can be a sign of pain or illness. Don't let that scare you because there can be a lot of other reasons for it, but it's definitely worth checking out!
If a vet check rules out any physical reason, I agree that crate training will help you a lot. Most healthy dogs will not mess in their crates unless they absolutely have to (they're sick and can't hold it, left alone too long--which it doesn't sound like you're doing, etc.).
You are handling the accidents correctly, just in case you needed reinforcement there.
Are you cleaning thoroughly with an enzymatic cleaner? If not, I would do that, particularly if he is going in the same spot each time. It may be that he's just found a new favorite spot, and getting rid of the odor (which you won't be able to smell, but he can) will help with that.
Basically, you'll have to re-housetrain him. Put him back on puppy lockdown--always supervised outside, and either tethered to you (or loose in an enclosed room where you can keep an eye on him) or in a crate if you can't watch him because you're asleep or out of the house. Praise him any time he eliminates outside (even urination, which doesn't sound like it's a problem in the house--but still, err on the side of rewarding him for it). Make it a big party if he actually poops out there. The good news is that it shouldn't take as long as initially housebreaking him.