That (flue gas temp) is indeed the right train of thought. The point is that the gases are already rather cool, and it is important to not loose too much more heat to avoid condensation of creosote (see also your cap pic). I believe that you want to keep it above 250 F. SO if you have a double walled pipe, that'll help with that.
It'll also (as a consequence of the temp) help with draft, making your stove function better. Remind us what the total length is, stovetop to cap? And what of that is (now) single wall inside? Any elbows and horizontal sections?
Single wall, straight shot from stove to cap. 7'6" stove to ceiling. About 1' in the ceiling. Then about 8'-10', probably closer to 8' shingle to cap.