TBH, I think you will struggle, particularly for a spur of the moment booking. I doubt that there is much of a market for genuine massages at that time of the night, so few if any catering for it.
In my experience, Thais and Hungarians give the best massages. So my recommendation would be to seek out one that opens late-ish (or offers over-nights), arranges her own bookings, and try and negotiate a late booking. You will probably need to arrange it well in advance. And you are more likely to succeed if the girl already knows you fairly well, so can trust you to be ultra-discreet and not arrive drunk or disturb the neighbours at that time of the night.
You did say that you were willing to pay extra, and you are probably going to have to pay escort rates at that time of night. So have you considered the likes of:
External Link/Members OnlyExternal Link/Members OnlyIf you scroll past the £500 an hour offerings (gulp!), these both claim to offer £130 an hour incall girls, which is not that much more than you would pay for an hour's massage plus extras in a salon.