Spitballing, there might be issues with having one at L1 or L3, but I'd love to have one at L4 and L5.
* L1 is between the Earth and Sun, so orienting the sun-shield toward the sun would result in the cold-side being heated by reflected light from the earth. It would also have access to the same patch of sky as at L2 for any given time of year, so there wouldn't be an advantage to having it at L1 as opposed to having a second telescope at L2.
* For L4 and L5, there's a ± 60 degree orbital offset relative to Earth. That gives access to a different part of the sky at any given time, but over the course of 6 months, both locations could access any part of the sky. There's a potential for time-correlation studies, where an observation at L2 and L4 are made at the same time, with the ~150 million km between them serving either as the base of a triangle for parallax measurements or as a known distance for measuring speed of travel between the
That said, I'm not sure how useful those would be. The parallax measurements could be done by ground-based telescopes simply by waiting a few months between measurements, and the speed of travel measurements would primarily provide a measurement of the index of refraction inside the heliosphere.
* L3 is opposite the Sun from the Earth. This would have the best baseline for parallax studies, but would have communication issues as any direct line of sight would be blocked by the sun. There would need to be relay satellites in order to communicate back to Earth, though previous L4 and L5 satellites might be able to serve this purpose.
Overall, I'd guess the biggest scientific bang for the buck would be to load up more telescopes at L2. The proposals for the first 12 months of observations totaled about 48 months of observation time [0], so most of the proposals were rejected. The biggest advancement I can see would be to simply have more telescope time overall.