If the towel rod is a Taymor, then you should be aware that your local hardware wholesaler will carry 3/4 inch square chrome plated Taymor towel rods in lengths up to 48 inches. You can use a stud finder to locate the studs in the wall.
(Press your rod supports onto your existing towel rod and measure the distance between the centers of the rod supports. Compare that measurement with the length of the rod itself. Once you know the distance between the studs centers, you'll then also know how long to cut the new towel rod so that it'll fit properly between the rod supports.)
If it's not a Taymor, and you can find the manufacturer of your towel rod on the internet, you could find out who their distributor in the US is, and maybe they could send you a longer matching towel rod that you could cut to length.
Mounting BOTH towel rod supports to the studs is always the best choice, but it usually also means you have to get a longer rod and cut it to length. Alternatively, buy two towel rod sets and cut the rods to about 16 inches each so that the rod ends can both still be screwed into studs. (Most houses have thier wall studs 16 inches on center, but some use 24 inch centers, so check your stud spacing with a stud finder first.)