Sorry, I haven't been online for a few days, didn't mean to keep you waiting.
First I would like to check the 'birdmouth' cut in the existing rafters to be sure the tails are not weakened too much. If the cuts are 1 1/2" deep or less you could install a 2 X 6 'fascia' board and hang the new rafters on that board with 3, 16d nails driven back through the 'fascia' at a slight angle into the new rafters (2 on one side and 1 on the other). This will require removing the soffit material for nailing access and you may find there is already a 2 X 6 'fascia' or 'sub-facia' ( if it is a sub, you will need to remove the 1 X 6 in the area you plan to build the patio roof). I said all that to tell you the best looking choice, to me, is your second method.