Here are a few:
The Med - volunteering is organized through the chaplain's office, I believe.
You'll be exposed to a lot here; it's one of the busiest trauma centers in the country, though you will probably be cleaning house at first.
The Church Health Center - I'm not sure how much clinical exposure you will get as a volunteer here, but it's a cool place...the largest health care clinic for the poor supported by a local inter-faith community. I volunteered at the CHC for a year and had some great experiences...it just depends on what is available and needed at the time. If you at all identify with medicine as a spiritual vocation then this is a great place to develop that...otherwise, it's still a good, compassionate place.
In the Hope and Healing center at the CHC you can volunteer to be an exersize partner for an elderly person...or a person who has a hard time excersizing...you just meet them at the CHC at the times the two of you work out so that they'll have somebody to walk with them around the indoor track. Sounds simple, but it keeps people who don't have much hope coming back...it saves lives.
Other non-medically related places to volunteer:
Streets Ministries - it was on the cover of the NY times. This guy built this place where inner-city kids could come hang out, learn to use computers, etc.
Hands on Memphis - this place may now be defunct...it's purpose was to coordinate people (volunteers) with needs in the community.
MIFA - Memphis Interfaith Association...an incredible organization in Memphis. They do tons for the city. You can get a Meals on Wheels route, which is something that is really cool to do...you'll see a side of Memphis that few people see. I'm sure they can hook you up with several other volunteer opportunities.
St. Jude's Children's Research Hospital - you can play with the kids being treated for cancer, donate platelets, etc. ...not for the faint of heart...it, I'm sure, will give you somethings to talk about when (and if) you have to write a personal statement.
Soup Kitchen at First Presbyterian church - for years this downtown church has served a hot meal every sunday afternoon. Different people come in to help pass out condiments and beverages, etc. It only takes a few hours...you'll get to know some of the regulars if you go there often. There was a homeless guy I worked with there that was in several national videos, etc. He spins I bicucle wheel on his head while he dances...anyway, it's pretty cool.
Memphis is a poor place...and there are a large number of faith groups and people with the time and desire to lend a helping hand, if only for a few hours a month. There are plenty of opportunities. After all, Memphis is one of the top three most cheritable cities in the country, per capita.
PM me if you have any more questions.