Surely the question depends on which airline you’re flying to and where, i.e. there’s no point saying GOT is good because you can’t fly to Asia from there. (I don’t think you’re talking about B2Bs since coming back to London wouldn’t help your mileage issue.)

If CX is your plan, they have lunchtime flights from AMS, ZRH, BRU, MXP, FCO, MAD (longer than LON) among others, which makes same day connection doable but a bit risky. They have evening flights from CDG and FRA so that might suit better if there is availability, although those airports are not otherwise known for their user-friendliness.

Or simply go a day or two before and simply take your pick.