I also flew the TRA 1W departure on Sunday afternoon out of EDNY.
I have just been in a similar position to your good self in the past and now know to have hard copies of all published SIDs in the cockpit as well as the G1000 ready to accept the SID and transition (which helps a schmuck like me a whole lot!) I also have the airport diagram including apron parking diagram to hand just in case I get cleared to somewhere not immediately obvious.
EDNY gave me an IFR clearance despite being very busy (and me having been unable to book one over the internet during the previous days) and Zurich gave me DCT TRA from the leg of the SID I was then following.
I always have the SIDs in front of me ready to shuffle through when I have copied the clearance and quickly brief it on the way to the runway holding point. After departure passing 1000ft I then engage the A/P and supervise it as it follows the SID legs (it flies them far more accurately than I can hand fly them) just to make sure it doesn't get any funny ideas.
Those ground controllers
can really catch a pilot out with the SID in the clearance so I always ask for clarification if I can't find the likely one.
My advice: If in doubt just ask, and be prepared with the SIDs in your hand.
SB