Stick a triple on, then you have your normal double and a lower range if needed, plus you can keep a close grouped cassette, I ride a 12-23 rear (12-13-14-15-16-17-18-19-21-23) and a 50-38-28, but I could change any of the ring, i.e. the 28 to 26 (not particularly nice combo due to having to go back quite a few gears, but gives a low 30" gear) or swap the 50 to a 52 (though I would need to move the FD up) or a 38-39-40 middle ring, or change the rear to a 12-27 (12-13-14-15-16-17-19-21-24-27) I think you can see why I changed to the 12-23, however the 50-38-26 12-27 worked pretty well apart from missing the 18th giving a range from I think it was 25" or so to 110",
However as you were asking about doubles, if changing to a compact (apart from the new 4 arm cranks) its the BCD (bolt circle diameter) of the chain rings that determine if its a normal or a compact a 39/53 is normally a 130mm BCD on a shimano, the compact is usually 110mm so the cranks have the holes in a different place, so going from a normal double or triple to a compact double or triple you most likely need new crank arms and chain rings the smallest inner ring on a 130mm BCD is 38th you can now see why I opted for a compact triple, I know the
FD can cope with a 24th large to small and does up to 12th inner to middle, middle to outer.
People tell me its not a proper bike, now I admit i am not fast average 15-16mph with an average 37-40ft/mile climb rate and round here I am never wanting for another gear (well apart from one hill where I nearly wished I had left the 26th inner on)