So couple things to consider
If you are trying to make a 1 night "reservation" at a campground in June that will be tough. Dont be discouraged however there are lots of campsites up the Poudre straight up out of Fort Collins. Some of the camp sites are a little rough as far as the ambiance associated with your fellow campers. The smaller sites are better but the big ones leave a little to be desire.
Check out this site as well as the National Forest Service site for the area
Camping on the Cache La Poudre River in Colorado
My favorite is Big Bend way up the canyon but it is small and right on the river and first come first serve. Also a ways further up is State Forest State Park with a lot of small lakes and a few high country rivers such as the North Michigan.This is a beautiful area and the odds are real likely you will fish amongst the company of Elk and Moose.
Check the flows since June is typically runoff so the Poudre down low runs real fast. You might need to venture down to the Big Thompson out of Loveland since it is a tailwater and flows will be more consistent.
Another wild option is head an hour North to Laramie and a little west for some secluded small stream fishing. There is a fly shop in Laramie that will tell you the creeks and campgrounds if you call them and ask.