I flew with Pritti a few weeks ago, cherry-head conure, Jet Blue airline. It might be a tight fit for the bird you are thinking of transporting. He flew in the cabin with me. I checked several different airlines with flights to my destination. Each one had a DIFFERENT size criteria for the pet carrier. In fact, one of my options was a connecting flight and each of the flights involved in that single trip had different size requirements. The first flight was a larger under-seat size than second. I would have been so screwed because they DID NOT tell me that when I phoned and the carrier I would have bought would not have fit second flight. I actually researched the aircraft on my own and found this out. Anyhow, the carriers that you would usually see cats and dogs in are not good options because most of them are 10.5" high or more -and- I found that 3 out of 5 aircraft I looked at had a 9" max height. I found a great carrier for Pritti. It's pretty spacious inside if the height is good. You can find it in the conure forum here, just search my name wannabeaparrot and you'll see the posts if you would like more info. It is a great carrier for a great price and worked out very well for Pritti.
Security was a bit of a pain, but that's the way it is. I wrote about that too. Be sure to leave an extra 30 minutes for checking through security. I had to go in the little room and take him out so they could take the carrier away and put it through the scanners without him in it. One TAS person told me that an African Grey had gotten loose and they were chasing after it for an hour, some weeks prior. But, all the others seemd totally stumped about how to handle screening a bird.
Call a bird breeder or large bird only store and ask abou the shipping. There's one in Deerfield Beach, Fl and another in Delray Beach, Fl - I won't list names because I don't know if it is permitted on this site. Just search for bird store in those areas. They both ship and know what airlines will do it. I recall that there are two airlines, Delta and possible American that ship the birds with special care. So maybe you can fly home and then have the bird show up the next day. It's lovely that you want to rescue him and not leave him/her behind.