I dont like a strap diagonally across my body because the strap forces my shirt to be close to my skin thus accentuating my buddhabelly and manboobs. Also, this reduces the cooling breezy ventilation provided by a loose off-body shirt, and causes sweat to wetten the shirt. I have a variety of fannypacks and beltpacs, and these somewhat hide under my shirt or jacket bottom. FWIW, many times I have been waved through entrance security at shopping malls and entertainment venues because the fannypack is not so obvious, whereas men with crossbody bags (and ladies with purses) are having the contents examined.

Sorry for not being more helpful to the OP request, but I just had to share some of my traveler wisdom!