An article in the Spokesman Review claims that the best solution for biting flies is a combination of Permethrin to treat clothing with Picaridin to treat skin and gear.
Supposedly the Picaridin is more effective than DEET against the biting flies and the Permethrin will add an additional layer of protection against insects such as mosquitos and ticks which may land on clothing.
Additionally, this Post from University Of Maine Cooperative Extension also states Picaradin and Permethrin are effective for flies.
I've also seen references to the efficacy of lemon eucalyptus oil for repelling biting flies, but with a less effective duration against Mosquitos.