You really need a per-channel FX send to do what you are asking but....
This is really hackish and may sound horrible, but you could try using the headphone jack as your source like you said if you:
1. Route the output of the reel back into an open channel on the mixer (Aux A-E for instance so you could assign it to any aux channel)
2. Select the channel you wish to echo with the headphone source select.
3. Once you have captured the sample, quickly switch the headphone source to the echo channel so it regenerates.
Never tried this so dont know what you might get. I'm guessing you will saturate the tape pretty quickly unless you are really careful with the headphone level. Also not sure if this is good for the headphone amp, but maybe someone else will comment on that.