Does anyone know if it's possible to unlock a US cell phone in Switzerland so I can use it with a Swiss sim card?

Probably. Contact the US service provider - they should do it for you - we just unlocked a verizion iphone 6, but verizon had to start the process. It was free.

Look on Internet, there are plenty of people selling software fixes for about $5 to $10

Try searching using the terms "jailBreak manufacturer model" and you should find what you are looking for.

I tried that but they won't unlock it until mid April or May. I'll be back in CH within the week and was hoping it can be done there quicker.

Ah, ok. I guess you have to have it two years or something. In that case, there are sevices that unlock the phone for a fee. It's not jailbreaking though that might be an option too. As sbrinz suggests, the internet is your best option.

I found a great site that charges by the type of phone. Some are $40-$50, amazing. This one is $5.99 for my phone but starts at $1.99. Code Unlock Store.

Thanks all!