Besides Red Cross, are there any other places that might be available on a deployment that Soldiers can receive volunteer hours?

I had a Soldier during my last deployment volunteer at the MWR when he had the time after his shift.

The answer to this question is really dependent on the location you are at. There are always opportunities to volunteer. Whether it is at an MWR, VFW, USO, or even something as simple as picking up trash. I will not pretend to know your deployment history, but I caution most on jumping head first into various volunteer opportunities upon arrival. Too many times I've seen personnel chase "kudos," bullets, or promotion points and end up exhausting themselves. Remember why you are deploying and the mission you are assigned to accomplish. Get into a solid routine the best you can. By the time you have that established, you will be more than aware of volunteer opportunities at the location you are at. If not, asking around or simply asking yourself "what can I do to enrich the lives of those around me?" can present a plethora of ideas. I commend you for wanting to volunteer. Being a servant to your community and those around you is noble. Volunteering during "off" time is also a great way to help pass the time and keep an active mind. But remember, there always needs to be a small part of you that puts YOU first in order to maintain the resilience necessary to operate in a deployed environment.

Ask around when you get settled in. Sometimes there are local opportunities like playing with the kids at the orphanage. Or just helping out somewhere else (maybe the hospital).