LJDRVR:
Unless you have great benefits with your carrier, it will be difficult for you to do something "ad hoc."
A little background: Most humanitarian aviation operators break down into two areas, NPOs and NGOs.
NPOs, or Non-Profit Organizations operate at minimal cost to the passenger/charterer, so don't plan on getting paid much. Also, most NPO avaition organizations are para-church or missionary aviation oriented. Most require donor support for your pay, i.e., you may have to raise it. All want a significant commitment from you operationally and spiritually. I don't know where are on that issue, but here are a few links:
1. Mission Aviation Fellowship-
http://www.maf.org. MAF is the world's largest missionary aviation organization.
2. JAARS -
http://www.jaars.org/aviation04/avia_home.shtml. Jaars (Jungle Aviation And Radio Service) is a transportation division Wycliffe Bible Translators. Their name is self explanatory.
3. AIM-Air -
http://www.aimair.org - Christian relief and development/Missionary Aviation organization located in Nairobi, Kenya. A division of Africa Inland Mission, Int'l. Africa's oldest missionary organization.
4. Samaritan's Purse -
http://www.samaritanspurse.org - Operated by Franklin Graham, Billy Graham's son. Relief and development, world wide. AIM-AIR and SP work together in East and Central Africa.
NGO's
Most noted is the UN. They have several branches like UNICEF, UN World Food Program, etc. I belief they contract with operators all over the world who have their own "121" or "135" operations. Perhaps some other guys on this forum who have flown for the UN could comment. I think the UN does have it's own aviation department. Doesn't sound like you could volunteer there.
AIR Serve:
http://www.airserve.org - Arguably the most popular humanitarian aviation operation in the world. They train you and they pay you.
Anyway, my point again is, I don't believe they will let you just come in and fly their aircraft without some kind of commitment (time-contract, etc.)
Angel flights are great if you want to do something state-side. Other than that, for what you are talking about, I can't think of much more.
Oh, I flew for NPO #3 for a total of three years (not at one time)on the African Continent. Some of the most rewarding experiences I HAVE EVER HAD came during those years. And the flying was spectacular.
T8