When I get this kind of a situation on a cloudy day in Kona, Hawaii sometimes I fix it with flash.
By using a off camera flash I set the flash to be 2-stops over the available light and the camera I underexposed by -2 stops.
While this made the photo much better the color just didn't pop on the background.
Here I added a CTO +1 and did a custom white balance for the flash on the model's face. I could have also just dialed the white balance to tungsten and been very close.
The last photo I put a CTB +1 on the flash and then did a custom white balance. Because the camera is compensating for the blue in the flash it added orange to the entire scene. Where the flash is hitting the model is now the proper color temperature.
So, which one do you like the best? Do you like just a flash added or would you add a blue or orange filter to change the background?