The White Mountain streams that penetrate deep enough into the WMNF to provide parallel trail hiking options tend to be limited to shallow, boulder-choked stuff that is generally best paddled when it's highest in early season. That said, these streams - such as the upper Saco below Crawford Notch - can be class 3 (or higher, such as on the Swift River) with plenty of potential to get "bandersnatched" in rock garden pour-offs. Happened to me last year, and I actually lost my raft (and pack and paddle) to the river, only to be recovered a week later via a lost-and-found ad!
I tend to think the best spots in the Whites for beginner and intermediate packrafting would be the same rivers used by the masses of summer canoeists and kayakers. The lower Saco, from Bartlett to Fryeburg, is the quintessential lazy river paddle by mid-summer, with some riffles and plenty of sandy take-outs on the meanders that would make for decent stealth camping prospects.
As an alternartive to hiking or biking back to the original put-in, taxi companies in Conway could easily accommodate a packraft and take-apart paddle.
More remote packrafting opportunities such as the Androscoggin River could also work for the intermediate paddler, with a hitchhike thrown in at the beginning or end of the trip. Paddling up-river is certainly possible, too, given a sub-2mph-ish current.
Also, some of us actually enjoy flatwater packrafting, especially for use in practicing ferrying, backpaddling, deep water re-entry, etc. The more interesting the shoreline, generally the more interesting the paddling, I find. And mid-lake obstacles like boulders and tree stumps are fun to maneuver around in circles, practicing finesse.
Swift River near Albany covered bridge, mid-May '07