What you need to do is simply expose for longer or with wider aperture or for a longer time (or some combination).

Let's have a look at your test :

shooting the same scene at 1/200 @f6.3 produced about 3 times as much contrast as shooting at 1/80 @f14

Now you omit ISO here, which is important ( was it auto ISO or something else ). I'll have to assume the same ISO.

Going from 1/200th to 1/80th means 2.5 stops extra (much more light) Going from f6.3 to f14 means about 1.2 stops less light.

So netting about 1.3 EV extra. A stop.

So this is consistent with what I said - it's about capturing more light in dull conditions. And that's your own experiment.

I don't know how you're measuring "3 times as much contrast" so I can't comment on that specific value.

On cloudy days (compared with those bright blue skies) there is a difference in the shape of the histogram you'll measure in each scene, even when you adjust for the basic level of light.

So when the scene is processed from RAW sensor data to a JPEG (which involves some quite complex remapping of values in a non-linear way), the result will be the (apparent) spreading out of the JPEG's histogram, but sometimes with the broadening of the curve (this will depend on the specific scene).

Because the curve is "stretched out" in that non-linear way you're going to see quite different contrast partly because of the stretching, partly because of the distribution of values being subtlety changed and mostly because the maximum intensity value is now shifted dramatically upward.

It's also worth noting that because cloudy scenes have different white balance from the sunny scene, this also affects your perception of brightness and contrast. Your brain is "tuned", in effect, to interpret sunny scenes as brighter than cloudy scenes. And that's ignoring that the tone curves (red, green and blue) are going to be different shapes as well.

Are there any other tricks I can use to stretch the contrast when shooting ?

This is where I'm going to suggest a technique I personally detest, but it's expose to the right.

It's important when using this technique to note that if you slavishly look at histograms you'll end up with a lot of under-exposed images. You have to use this technique sensibly and expose for the main things you want in the scene. Some things will get blown sometimes doing this.

Personally I have never found a better method that simply shooting RAW and doing a careful conversion later. You need to expose well for the scene, but in practice most cameras have very good metering these days and it's a matter of watching out for situations that need a little exposure compensation.

I know you want tips for when you shoot, but that really boils down to getting exposure settings right and that means learning to use your metering system and exposure compensation.