Is there an equivalent to heat index (temp & humidity), which also includes exposure to direct sun, perhaps as given by the commonly reported "UV index" shown on weather sites? In other words, temp measurements are always given for "in the shade," but direct sun & UV exposure is dramatically more meaningful & potentially dangerous to living organisms (especially humans).
@BenCrowell's comment is right. This was going to be a comment agreeing with his, but grew.
I think the best single number you could do is some sort of total sun risk index, which is of limited value as mitigations for the different risks are different (shade is good for both, clothing can increase heat retention while protecting from UV, extra water is needed for heat but does nothing for UV, and sunscreen does nothing for heat).
Dropping the UV consideration, you'd have an enhanced heat index risk including solar gain. The problem is that the amount of heating from direct sunlight depends strongly on clothing- and even skin-colour, and wind offsets the heating to some extent, though if sweat evaporates fast you might not realise how much you're sweating. Still, a solar heat risk index would be useful. It would be based on certain assumptions about exertion though, and you can't arbitrarily limit your exertion (e.g. cycle slower uphill, you fall off, paddle gently all the time on white water, you lose control - but it's hot in a drysuit in the Alps in summer, at least above the chilly water). But the risk index and the mitigations are too closely linked: stopping in every patch of shade for a good drink reduces your average solar gain, thus the index changes.
Overall I reckon you should risk assess heat and UV separately -- and cold, as I hint at in my kayaking example, but also if the area is prone to sudden storms, as many mountain areas are even in the height of summer; deserts can get below freezing if you're out overnight. When I say "risk assess" I don't necessarily mean a formal process as many of us have to do in work, but the informal rolling risk assessment that any leader needs to be thinking of -- and this isn't heavily based on numbers (it would be quite different if a member of the party was already dehydrated or had to cover up because they'd already got sunburnt)
Via the NWS:
The WetBulb Globe Temperature (WBGT) is a measure of the heat stress in direct sunlight, which takes into account: temperature, humidity, wind speed, sun angle and cloud cover (solar radiation). This differs from the heat index, which takes into consideration temperature and humidity and is calculated for shady areas. If you work or exercise in direct sunlight, this is a good element to monitor. Military agencies, OSHA and many nations use the WBGT as a guide to managing workload in direct sunlight.
More specifically a wet bulb thermometer has a moistened sleeve over the bulb. Combining the readings of a wet bulb, dry bulb, and black globe thermometer yields an apparent temperature which takes into account all of the above factors. (.7wet + .2black + .1dry for the curious)
Parenthetically AccuWeather offers "RealFeel" which is an apparent temperature which claims to include radiation. Since the formula is proprietary and US weather stations don't measure radiation directly it's most likely using an estimate based on location and sky cover. Buying a wet bulb thermometer is a much better solution.