I was pretty sure they were different words in the Torah ... גוֹזָל - gozal ... (young) pigeon ... e.g. Genesis 15:9 ... KJV, of course ... on pigeon in TaNaKh: http://quod.lib.umich.edu/cgi/k/kjv/kjv-idx?type=simple=Long=pigeon=Old+Testament=First+100
Gozal is a general term for a young bird, specifically nestlings, perhaps. תנ"ך might use it without specifying type of bird, sometimes because it is a known type, sometimes because it didn't matter.
Efroach אֶפְרוֹחַ is used for chicks.
Yona is dove or pigeon. (anecdote: in the "worms" game, there is a version where "_pig_eon bombs" were translated into "flying pigs" - "חזיר מעופף". I guess someone was very tired down there)
Perhaps you are thinking of the Passenger Pigeon, which sadly is an extinct species. However, feral and domestic pigeons (including "Carrier" or "Homing" pigeons) are descended from Rock Doves, which have a natural homing ability. They're more likely to be kept for racing than for carrying messages these days, but are definitely not extinct :-)
Thank you for the thought, Helen, but I had to look it up to be sure. See https://en.wikipedia.org/wiki/Homing_pigeon. The pictures here look like the rock doves I see outside my window in Yerushalayim. It's clear from this entry that these are the ones that were used for messaging back in the day.
Unfortunately, you are correct about passenger pigeons being extinct. You can see more about them here: https://en.wikipedia.org/wiki/Passenger_pigeon. Believe it or not, we have doves that look a lot like these here too, and they are my favorite (they don't fly nearly as fast or as high - they fly just as much as they need to in order to get to a tree branch.). No reference to message or letter appear in this entry.
Nice try, though. ;-)