I agree with everyone else that Harold Godwinson would almost certainly not want an overseas adventure. I imagine the cost of two wars would be enough to convince him of the need for rest (and potential attacks from Denmark and Scotland, rebellions etc.)
I do have another question however. What role would the Witenagemot have in... for want of a better term, authorising a new war even if Harold wanted revenge and conquest of some description?
I know they aren't a parliament or anything like that, but I imagine the ealdormen and thegns would be very resistive to further wars of conquest.
I agree, at best there might be raids, but certainly no invasion or even conquest; like others said it would be more interesting for them to start and influence any Norman civil war, which might arise after the death of duke William.