I'm saying contact the second party company (first party is who you work for) and fake an influence on your employer, the first party company. Imply that you can impact the second party company's business (because you don't like the 3rd party) by convincing your employer to go with another firm that has better 3rd party care.

Since you're just changing jobs now, I don't know that that will work, sounds like you just got a job with a company who buys lousy insurance and, since you just started working there, don't have any leverage to use in fact.

I had great health insurance but my company wanted me to work part-time to save money. However, because of my past history with the company and good service to them, I was able to negotiate so I kept my insurance, as if I were still full-time.