That should be a good time. While we can get rain and cold any time of the year, by later in the summer, it can get quite hot and there is not much in the way of water after you get past Harrison until you get closer to Kellogg.
One bit of advice, enjoy the scenery. I have found it quite easy to just get into the ride and pedal past all kinds of things that would be better enjoyed by stopping and just taking it in. If you manage to get started early in the day, you will probably come across lots of wildlife on the trail, so you will want to be ready for the unexpected. I've had to work my way around all kinds of animals from turkeys to moose.