Make a plan for the both of you for Wednesday (don't really, but he won't know that)
When you go home tonight, tell him about the plan
When you suggest it, his body language and how he answers you will tell you everything you need to know.
Also, if he says hes 'going to the gym' can you turn up at the gym, possibly with some fabricated excuse? Or do you belong to the same gym?
I understand you wanting a little bit more proof before confronting.
With what you have at the moment, it's to easy for him to weasel his way out of it and make excuses:
''Why were you telling her your room no. late at night?''
Him: ''We wanted to all go down to breakfast together''
''Why are you arranging to meet her on Wednesday?''
Him: ''She has a work problem that only I can help her with''
''Why have you never mentioned this person''
''I wasn't doing anything wrong so there was no need''
and then he will say:
''Why were you snooping on my phone'' and make out you're jealous and insecure and that you're the one with the problem.
If you get a little bit more proof he won't be able to worm his way out so easily.