Five years ago, I wrote an article for this blog called "Dressing for the Weather". In that piece, I covered use of linen, types of garb and some other tips based on my own experiences over the years. Ability to cope with the weather, is, of course, a very personal issue and while some can parade around in full Elizabethan at Pennsic, others would find their comfort too compromised were they to dress in that manner. Because this is the SCA, we allow for a range of options, and strict period attire is not required. There are still plenty of ways to try to fend off that humidity, with a passible historic guise.
One way to do this is simple single layer garb. Many people opt for Roman or Green in the summer months for this very reason. A chiton or peplos can be a brilliant way of combatting the elements.
Other options include Bog-dresses (which is also a peplos). This garment can be voluminous for a very historic look (and in linen, with no belt, that can be exceedingly comfortable in the muggy Pennsic afternoons). Finds from Huldremose, Zweeloo and Hammerum all give us peplos garments that are a tube of cloth that is pinned at the shoulders. Huldremose has a fold at the top that creates a flap while the other two have the pins at the edge. The Hammerum dress is also short, which actually allows you to make this garment with less fabric (and less bunching under the arms and at the waist) than the Huldremose version.
For something still passible, but that uses minimal cloth and reduces bunching, I have a document on creating a "less fabric" version of a bog dress, that uses pleats at the top for a better drape. Without the pleats it is actually pretty comparable to a "Germanic Barbarian Dress" from the Hungarian Museum. You can find that item here: http://awanderingelf.weebly.com/blog-my-journey/sca-standards-the-bog-dress