Where the garage gable peak meets the house gable, extend that line across the house roof, I would stop short of the far overhang about four feet. Then set the pitch you want for that new roof line out from the house roof. Either chose a pitch you find the best looking and where it meet the height of the overhang of the house and garage is where the new roof ends. Or extend the new roof out to the garage overhang and that is your pitch. However, I think that pitch will be too flat. I'd stop the new roof about four to eight feet short of the garage roof overhang and it will provide the best balance of roof size, proportion and pitch.
The outermost corner of the new roof will have to be supported by a post or posts set on a concrete post down to the depth of the house foundation. In Virginia that may be two to four feet deep.