First, you're dealing with "shaft" lighting, meaning the light is coming straight down through the trees on to your model. That is the least flattering light there is, and is why most photographers prefer, when possible, to shoot in early morning or late in the day (my choice) so that the sun is lower in the sky and comes at a more level angle, either front lit, side lit, or back lit. Even during the day, if the sun as at SOME angle other than straight up, you can have the sun used as a back light/hair light, with some bounce fill from the front. The other problem you have is all the greenery bouncing greenish light onto the model. \
The best way to fix both problems, (and add nice element to the photos) would be to use some sort of overhead scrim - a white "silk", in reality, something like parachute material. You can buy these from pro photo suppliers. I have a Matthews 6'x6' frame with a reflector surface, and also a white nylon fabric. Just put it over the top of the model (on "C" stands) and the light will be diffused and soft. Then, get some bounce cards (preferabley white for this use) and bounce some sunlight directly onto the model. Don't place them on the ground, get them up about 3' off the ground otherwise you'll get shadows under the model's eyes.
All of this will do three things: Give you a soft top light, some fill on the model to give you some punch and eliminate the green color cast from the trees, and (as you will find when you open up the f-stop to compensate for the additional exposure you'll need) you will make the foliage a little more out of focus, and lighter in color, which will be more dream-like and attractive.
Notice I never mentioned your strobe. Personally, I wouldn't use it for all the reasons mentioned, but you could replace the bounce cards with the flash (on an umbrella preferably) as long as you don't overpower the scene. Keep it just under the exposure of the top light. Also, if you don't have a Matthews silk, you can use any white nylon fabric held above her head, or go to Costco and buy one of those white 10'x10' canopies for $200.
There's one other way I just thought of. Take your model to the edge of the woods, so you can back up, wait for the sun to go down behind her a little bit, and use the open space around you to place a large reflector (or your flash). The point is NOT to have the shaft of light coming down if you can avoid it. BTW, as you get closer to winter, the sun spends more of the day NOT being overhead, depending on where you live. For example, here in LA, in the winter, I can shoot outside all day because the sun never gets direcfly overhead. In summer I avoid shooting mid-day at all costs. Here's an example of both those things. We shot this on a mountain side, but there was a 25' clearing behind me, and we shot in February last year, about 3 hours before sunset. The only light modifier I used was that 6'x6' reflector off to my left (model right) and the sun over her shoulder.
Andy Pearlman Studio