if you want to house train without a crate, there's only one way:
supervise your puppy every second of the day.
accidents happen when you can't act on the first sign of the puppy having to go. and every single accident you allow to happen means a setback overall. it's not the dog's fault, they don't know better but are trying to learn to please you: including not soiling in improper places.
so keep your eyes on your dog. if you need your hands free, put him on a leash and tie it to your waist. feed, water and take him out to potty on a tight schedule. don't turn your back on him for a second while giving him free run of the room, and don't give him free run of the entire house until he's at least mostly reliable.
all this is easier with a crate or x-pen, since you can pop the dog in it when you can't supervise, even if it is just for 5 minutes while you are on the phone with a friend.