The thermoregulation experiment suggested that cooling the hands is a relatively efficient way to cool the whole body. I don't think many people feel like their hands get too hot while they exercise, but there are apparently gains in endurance when exercisers keep something cold on their hands. These gains are most likely due to the body taking longer to overheat.

You could find out whether this works for you by timing how long something takes to produce noticeable overheating or sweat, then timing the same thing on a later day in very similar conditions, holding ice packs or something like that in your hands.

If you don't want to induce extra overheating or sweat for the sake of the experiment, you could try holding something cold while doing something you have to do anyway (e.g. sometimes I have to go outside on a hot day). That way the worst likely outcome is not much worse than before unless you really hate holding cold things.