Can I get a refund on the e-file fee that I paid? My return was rejected. I ended up having someone else do my taxes.

We cannot tell what software you used.  Do you mean you used online software or that you used the CD/download?   If you used the online program you have to pay for each return---and if you got far enough to e-file, yes, you will be charged even if the return was rejected.   With online you pay the same fee whether you e-file or mail it in an envelope.  Rejected returns can be fixed and re-filed or they can be printed and mailed.

if you used the CD/download and paid the state e-file fee and the return was rejected, you can ask customer support for a refund of the state e-file fee.

To call TurboTax customer support

https://ttlc.intuit.com/questions/1899263-what-is-the-turbotax-phone-number

Customer support is available from 5 a.m. to 5 p.m. Pacific time Monday-Saturday.

No one in the user forum can resolve a billing issue.  If you have a question about your TurboTax fees or billing, make sure you use the word “billing” in your request for help.  Do not use the word “refund.”