Im not sure you can attach to an amendment...what are you attaching?  Are you sure its necessary? If youre trying to submit some kind of required back up documentation, you should really send it by mail. 

EF 1040Xs dont get processed any quicker than a paper one, they print them out and they go in the stack like any other, you just save on the mailing time.

PS:  I had a client get  a letter last week, we EF a 1040X in March, the letter said Thank you for your 1040X received June 7th, but you forgot to sign the return, please send a new copy of the 1040X with a signature...so they cant even tell the difference between whats been mailed and whats been EF'd.