If I'm shooting midday on a cloudless day, say, under a gazebo, highway overpass, etc., what would that color temperature be. I know it's higher than 5600K.

It depends, but I would expect it to be in the neighborhood of 6500K.

When you're outdoors, there's usually two main sources of light: the sun and the sky.

I can't remember figures for both, but sky light will be very high color temperature, around 9000K.

2- Kelvins actually aren't a great way of describing a color, as they work only for black body radiators (things that emit light when heated up). In real world situations, you have to deal with things like pollution or light sources that aren't black body radiators. i.e. Fluorescents are typically greener than the equivalent black body radiator / correlated color temperature.

3- Bryan, what exactly are you trying to do?

Bryan,
Outdoors bright sunlight 5600K
Light Shade or Light Clouds 6300K
Dark Shade or very Cloudy 7200K on up to 10000k or more

Of course these numbers are useful only as a reference. The farther up the Color Temperature scale you go the less sensitive video cameras are to the color differences. If you white balance and the number says 7000K in one place and 8000K in another there is probably not much difference in the look even though there is 1000K difference in the numbers because the blue gain is already turned down very low and the red is being cranked up as high as it can go. Unless there is some other color cast to the location which the camera sees. As an aside When Sony let you set the Color Preset for Outdoors on their cameras I would always choose 6300K because that worked for light shade and Direct Sun looked warm but nice.
At the lower end of the scale there is a big difference between numbers which are seperated by only 200k

At the lower end of the scale there is a big difference between numbers which are seperated by only 200k
Just try using work lights (~2800K) instead of film hot lights (3200K) to see for yourself.