What he said.
However, CJ hwy stops at the Beartooth Hwy. Turn left, go to Cooke City and the NE entrance/exit to Yellowstone. However, if you turn right at the beartooth you get some fantastic views and a drive into Red Lodge.
Two weeks ago, I did from the campground 20 miles east of Cody, through Cody to Red Lodge, and from Red Lodge the Beartooth Hwy to CJ hwy and then back into Cody and to the campground. Took a day and I stopped quite often to take photos and in the switchbacks, on the beartooth, let the dog bury herself in the snow for about a half hour.
Last four photos are from the Beartooth, and the last photo (Jim Smith's Peak) is only a few miles from the CJ parkway.
Is it a worthwhile trip? Hell yes!