I have run North Platte for almost 30 years from levels of 4700 to 300. Usually by late July it is in the low hundreds, but not this year. It will likely be mid August before it hits that. I have run a 14 foot raft with 4 people in it at flows of 500. It wasn't great, but we got down and the fishing was fantastic. At 300 you can take Kayaks or mini me's but that same 14 foot raft needed to be dragged in many places. 1000 to 2500 are the best flows.
Low flows (~500 cfs at Northgate Gage) in small boats can be a lot of fun, although technical, and the fishing and weather are usually good that time of year (late June and July or August). Sometimes we will do 3-day trips and take our time fishing and hiking travelling to Pickeroon. The river is accessible on the west side of the river at Pickeroon, but it is a high clearance 4WD road, and not good if it is wet or has snow. High flows above 3,000 can be very PUSHY, with few eddies and no way to scout rapids in rafts. Kayaks are easier but can still be pushy at flows > 3,000. There are several places to camp in the main canyon above Narrow Falls and a lot places to camp downstream of 6-mile. At flows > 1,000 cfs it is easy to make a 3-day trip going from put-in in Colorado to Saratoga, and there is a NICE hotsprings in Saratoga (free). Fishing can be outstanding. If you have small kids or people that don't want to run the bigger whitewater, they can hike in 0.25 mile at 6-mile on the first day and join you for the rest of the trip. This is a nice multi-day trip without need for permit or the crowds. If you travel down below Pickeroon make sure you know the land ownership and have a good map. There is private land down there and they are not happy with people trespassing, which includes wading and getting out to take a leak. I think the thing that keeps the Platte from becoming a more regular multiday float is that it does have significant Private lands starting mid to late day two on a three day trip from Routt access in Colorado to Treasure Island or Saratoga in Wyoming. But there are plenty of nice places to camp if you do it. There is a good map for the North Platte that is available at stores in Laramie, Walden, or Saratoga and the maps do a good job of marking the private lands. Take the private land seriously, especially the first big ranch you come to. They really do not want you there, but they can't say much of you are just floating through. Do not stand or wade in the river or on the bank--they will chew you out and could press trespass charges.