The Red Creek trail gets very little use; the first two miles had many sections that were almost overgrown with waist- to head-high vegetation. Riding a mountain bike would not be an enjoyable proposition here. When you could see, the views were impressive.
After about 1.5 miles, the trail veers northeast up a side canyon, and eventually gains a ridge after about two miles. Getting up into the grass and sage was a welcome relief after the dense vegetation down in the canyon bottom. The scenery also improved considerably.
In Rebecca Woods' book Jackson Hole Hikes, her commentary on this trail says that the map shows a trail heading north toward Red Peak at the 3.1 mile mark, but she didn't see any sign of it on the ground. I didn't either, some 15 years later, and blithely followed the obvious trail that kept heading east toward Dry Fork Canyon. The views were gorgeous, but the trail wasn't taking me where I wanted to go.
By the time I realized that there wasn't a junction with a trail up to Red Peak, I was about a kilometer northeast of where the junction should have been. No matter though, there was another gentle ridge that headed toward Red Peak, and the terrain was a mix of grass and low sage that made for easy walking. The weather was warm, winds were almost calm, and in some places the wildflowers were a spectacular carpet of color.
After about a kilometer of easy off-trail travel I gained the main ridge leading up to Red Peak, and found the trail at that point.
I started fairly late; didn't leave the trailhead until 1:45 pm. I figured that 6pm was going to be my turn-around time so that I wouldn't be hiking out in the dark. Even with taking the long way around, I made it to the summit about 5:30pm.
The summit of Red Peak is 9736', which is a respectable hike from the trailhead at 5800'. The temperature was in the upper 70s, the winds were almost calm; the only minor issue was the smoky haze that spoiled the gorgeous 360-degree views here. I didn't see another person for the entire hike, although there was a band of sheep being herded along the next ridge to the northwest. The plume of dust in this picture (looking northwest) is from the sheep.
The views were spectacular in every direction. This is looking west to Deadhorse Peak; the map shows a trail that loops over to Deadhorse Peak and then back down Little Red Creek. I could see the trail on the ridge heading west, but I decided to save that adventure for another day when I wasn't short on daylight.
Looking to the north, the Tetons were just barely visible through the haze. I took a picture but they aren't visible in it. Oh well..
I spent about 15 minutes on top of the peak, had a snack and savored the views. Then it was time to head down. I followed the trail down the ridgeline -- it was easy to follow in some places, and totally obscured in others. Eventually I came to the cross trail that I had ascended on, right at an unusual rock where I had stopped for a short break on the way up. The trail up the ridge was totally invisible at that point, no wonder I'd missed it. So, if you're headed to Red Peak and you crest a low ridge and see these unusual rocks, turn left!
It was about six miles back to the car, and I covered it in about two hours; it was of course much easier going downhill! All in all, it took me just over six hours for the round trip. I didn't see another soul the entire route, the weather was perfect, the views were spectacular... wow.