Hi Cote, I hope you're doing OK. I just wanted to add a different slant on this perhaps which might help?
My DH and I don't really interrogate each other's social lives very rigorously so he could I suppose say 'I'm going to meet XX in the pub' and in reality be going off to meet a woman somewhere, and I could do the same. But basically we trust each other and so I've never felt the need to probe any further - never checked his email or phone and he doens't check mine (neither of us use FB for which I am very grateful!).
However I think I am probably guilty of what your DH is doing which makes me think he might be an innocent in this (I hope so). I love my DH dearly and we are extremely happy, I would never cheat on him. But I am in touch with an ex-boyfriend from uni days who I see very occasionally, maybe once or twice a year. When we meet we have a drink, perhaps have dinner, flirt a little bit and reminisce etc. as well as talking about our partners, work, mutual friends. Nothing ever happens. I am always home by about 10.30pm having had a lovely evening with an old friend and having been flirted with very gently in a harmless sort of way by someone who remembers me as young, thin and sexy.
If DH asks I say 'Oh I was out with XX from uni' and he never asks anything more. If we were to be brutally honest about it I would have to say 'XX and I had a relationship years ago, there's still a little bit of frisson between us' but this wouldn't serve any purpose as it would hugely upset my DH and it would be massively overstating the attraction I have with my ex.
Reading your OP I think your DH may have a similar situation with this woman. I do do hope so.