Is it worth it to apply to companies not listed as sponsors?

I'm currently finishing my MA and hoping to switch from Tier 4 to Tier 2 before it expires (so that it's unrestricted).

I had been focusing my search solely on companies that are already approved sponsors, but the lack of openings in companies in my field has been getting me down. Is it at all worth it to apply to companies with the hope that they will then apply for sponsorship? Does any one have any success stories doing this? FWIW I still have about 6 months left on my Tier 4, so would be able to do a sort of trial period before the company commits to sponsorship.

It doesn't hurt to apply.  You never know.

But I believe it takes about six months for a new company to obtain a sponsorship license (not to mention the cost), so it will be an uphill battle (if they have a position that needs to be filled quickly).  Also you may struggle within the time left on your current visa.

I was in the same position as you last summer, finishing my MA and wanted to switch to Tier 2.  I went to a few job fairs and was able to find a company who was willing to go through the process of attaining a license.  I honestly walked around and just asked if they would provide sponsorship. You may not find the job of your dreams, as I did not, but Im here now with plenty of time to find the job of my dreams.

Don't get discouraged, finding sponsorship is not easy, I have been looking for many months now and nothing has come up in terms of an employer being able, or wanted to sponsor. You may have to come to terms with if you want to stay in England, you may not be in a career that you were intending on.

There is a job fair coming up in London I recommend going too if you are able.
http://summer.londongradfair.co.uk

Good Luck!

Thanks for the encouragement. It's good timing as I just got rejected from a job I was so hopeful about (there was an alum I've had contact with working in the same department!) without even an interview 

Unfortunately I'm taking a trip home next week, so can't go to that job fair. But I'll keep my eyes peeled for any others.