Would you actually get a lower rate? Stretching your term to 20 years seems pointless if your goal is to pay it off sooner.
I'm not worried about the rate. The longer term would lower the monthly obligation, allowing me to put that difference towards the 13k loan to pay it off faster and then swinging that payment to the 22k loan.
To put numbers to it, I think what the OP is proposing is this:
Assuming a 6.5% rate on a balance of $22,000 (roughly what most SLs are currently)
A 10yr note would have monthly payments of $249.81
A 20yr note would have monthly payments of $164.03
If the OP wanted to put $500 towards that note the OP would pay the note off in 4 years 4 months for a total cost of $25,198.55 regardless of whether it was a 10yr or 20yr note.
The OP gets no "advantage" if it was just this one note. But s/he has two notes, including a smaller $13k note. In this case, he/she is going to put more of the initial overpyament towards the $13k note, allowing the OP to discharge this loan a few montht faster.
AT least, that's how I am understanding this. It's basically a version of the snowball method.