IMO you'd want to unsolder the original wire and redo the whole thing. Speaker wire carries current, that's it. Lower gauge (thicker wire/more strands), higher capacity for current. As soon as you introduce a thinner wire, you're reducing the maximum amount of current that can go down that "line". The only analogy I can come up with for this is imagine you work for a shipping company, and you fill your truck full of arcade machines. You travel 3,000 miles across the country and your truck breaks down with only 5 miles to go. Fortunately, a nice fellow trucker picks you up, but only has room for 80% of your shipment, so you end up having to leave 2 or 3 machines on the side of the road. At the end of the day, it didn't matter how far you went in your big truck (fat wire) because before you got to your destination, you had to drop cargo to accommodate the smaller truck (thin wire) that carried you the rest of the way. Now, if you're not maxing out your new wire, and are pushing enough current that the old wire can handle, you don't have a problem at all. However, the only upgraded performance you'll see is if the new wire is twisted, whereas the old wire is straight, you'll benefit from some noise reduction as it's harder for DC current from nearby electronics to leak into the twisted speaker wire.
Then again, 90% of what I know about sound comes from working on cars and not indoor stuff, so this could all not apply at all, lol.