I just called them a couple of days ago to ask this same question. The answer I got was that they are only allowing SWA employees to jumpseat. This is the new post 9-11 policy. The person I spoke with wasn't sure if this policy was temporary or permanent.
The person you talked to, while technically correct, was way off the mark. Only SWA pilots may OCCUPY a cockpit jump seat. However, 121 pilots may be offered a cabin seat with jump seat authority. I don't know if 135 pilots are universally allowed on via this authority, but I believe that certain specified 135 carriers are.
I bet if you asked a pilot from SWA, they would tell you its up to the Captain's discretion as to if they will accept Part 135 jumpseaters. I jumpseated several times when I was a Dispatcher and remember the flight crew saying they are willing to accept most pilots and dispatchers with a valid company ID, as long as the Captain approves it. Ofcourse, this was over a year ago, but I'm willing to bet the policy is somewhat the same.
SWA has not been allowing Part 135 jumpseaters since the September jumpseat fiasco.
According to the SWAPA website in a letter from 10/19, they are at least attempting to get AirNet and Ameriflight back on board. Those companies serve many of the "out of the way" airports that SWA serves.
It may be worth a try, but I wouldn't count on getting a seat. The 135 policy is pretty well known amongst the pilot group- not that we are happy about it, we're trying to get things fixed.