R.B.
I have had a similar experience. Have you ever confronted him as to why he lies about the little things? It sounds as though he does not trust you either if he feels the need to sneak around. You should start an open dialog with him about your expectations of each other. Tell him how you feel about the other woman and explain that the sneaking around does not improve your view of her. If anything, it makes things worse. Having said that, your snooping might prompt him to lie because he wants to avoid the confrontations. Does that make sense? I am on your side here.
If you truly want to be with this man, you do have to stop snooping. It will only make you crazy. I could easily assume the worst about my own husband. I have to empty his pockets to do laundry and I know he eats lunch out a lot, but I trust him to do the right thing. If your boyfriend is cheating, it will come out somehow. In the meantime, do your sanity a favor and let things be. Otherwise you will never be able to trust him.
You also might try to start some family time. I do not know what your schedule is like, but make it the same time every week. Remind him of the 2 most important women in his life and show him that he can have fun with his family instead of going out. Hope this helps. Feel free to message me if you want to talk further. Take care.