Great news! So is the closet in the living room? And the cage is inside the closet? And you think the hamster will now go into the cage where there's food and water I'm guessing. At least you know where he is, and he has access to food and water. However they sleep during the day so he may already be asleep curled up in cosy clothing or something, and not go in the cage until later. But at least he has access to food and water and you know where he is! Keep that closet door closed!
The tricky bit now is when you open the closet again. In case he runs out again! He's had a big taste of freedom now. So I would keep the closet door closed and listen out for any noise. If you hear him running in his wheel then you know he's in the cage and then you can open the closet door and quicky close the cage door.
I'm wondering if he's a roborovski dwarf hamster if he's really tiny (and they are very fast!).
This is a robo - they have quite a distinctive face and markings - they look like they have "eyebrows" with their markings (unless it's a white face robo).
View attachment 10732
This is a Russian dwarf hamster
View attachment 10733