Detroit should trade with either Chicago or New York if they want to move down. Drop to either 8 or 9, ask for 26, 27, or 28th as well, and to sweeten the pot a bit Detroit should offer their 67th to start but would most likely end up at the 36th pick.
Det/Chicago - Det gets 8th and 27th. Chicago gets 6th and 36.
Det/New York - Det gets 9th and 26th. Rangers get 6th and 36th.
Detroit gets 3 first round picks and it would truly only cost them a 2nd to do so in this case. I would bet that they are interested in a few guys and one of them has to be available at 8th or 9th.