There have been many wise words of advice already said on this post. Although they go against what you feel you really want, please do consider them all as many point to a similar narrative. Now is the time to cut to the chase and see if this relationship has any chance of lasting for the long haul. The simple answer is what many have already said - end it now and move on to a healthier, less complicated relationship before you get too deep, lose your self esteem and mess up your mental health. However, if there is at least one tiny chance of this working, you have to communicate together much much better. Now and forever...
Perhaps 'confronting' him is the wrong word, but, you 100% need to discuss it with him in a neutral environment when you both have a level clear head. No raised voices, no alchohol, no sex... his reaction (not just his words) will speak volumes.
From my experience, honesty really is the best policy. You can't spent the next 5-10 years worried about him being upto things and communicating secretly with other people as this will destroy your self esteem and cause you mental health issues. Although you've already fallen for him, with it being early in the relationship, now is a good time to explain to him your feelings and how much you are on edge thinking about this all the time. Bear in mind you are already worrying about this and you've only been together this time around for a short period and you ended your previous relationship with him for similar issues.
Yes, you snooped, but ultimately you found something straight away that could be a red flag. If this relationship is going to work long term, then you both need to begin communicating fully, openly, and honestly with an enhanced level of maturity. Otherwise your future mental health is at risk and this relationship is doomed.
At the end of the day, if this guy is honourable, there should be no message he sends anyone that he is not comfortable with you reading, especially if it is one to a woman. This obviously works the other way around too.
He had past issues when he was young and dumb and he was flirty with girls etc... well this new relationship with him isn't going to work if he gets up to his old tricks again, nor if he still holds feelings for the mother of his child. She will be around forever by the way, so you have to work with this scenario if this relationship is ever going to work. For your own sanity, you need to have a fully honest conversation with him around where his loyalties lay. Perhaps you should meet the other woman and be introduced as his partner and you both look after the child from time to time? Afterall, she almost certainly will meet another guy at some point so the door will swing both ways. He also needs to accept that. If he, she or you cannot handle this forever, then this relationship has already failed.
If he is the man you hope him to be, these conversations will be productive, will lift a weight off of your shoulders and you will build a deeper connection with each other, more than you ever have before. Then, honest, open, regular communication should form part of your long term happy and healthy relationship.
If the conversation goes south or you argue or he gets angry and defensive at you snooping (diversionary tactic), then I do not think he is the partner for you long term. He already has several failed relationships behind him and has a child (one, that he knows of) that he has left for the mother to look after. He already has you second guessing his actions/motives. His reputation is already majorly questionable and so far (I assume) he hasn't done anything to give himself credibility as long term partner material. I think you know in your heart of hearts what is right for you long term. Please tread carefully and best of luck OP.