I would say adding a steel header into the masonry wall and putting in an opening for a walk-thru to a sort of lean-to addition on the side or end would be about as far as I'd want to go with an assumption of "reasonable cost". Trying to take out an entire wall, either the sidewall or endwall (or both!) to go double-deep would be very expensive in my opinion. Not impossible of course, but a lot more than "tacking" the basic square footage on to the existing perimeter.
Edit: to add on it will need a foundation of its own, please update your location in User Profile, but consistent with local frost line requirements.