There are a few things that influence your body's ability to regulate heat. Nutrition, clothing, behavior, and practice are the big ones.
Nutrition is important as far as making sure you're properly hydrated and are replete with electrolytes. It's possible to become dehydrated even while drinking enough water. To fix this, make sure you consume salts such as sodium chloride, potassium chloride, or others you can read about here https://en.wikipedia.org/wiki/Electrolyte#Physiological_importance.
Clothing is important as well, and is easy to do incorrectly if you're unfamiliar with how. You've mentioned you're in a forest, which means you don't have to worry about providing your own shade. Focus on wearing lightweight clothing which allows ventilation. Generally, do not wear cotton. It will absorb several times its weight in water, allowing chafing while preventing your sweat from properly evaporating off your skin. The sweat must evaporate from your skin or a layer of clothing near your skin in order to cool you off; this is called evaporative cooling. Underclothing should also not be cotton, again due to chafing and because your groin is just about the warmest part of your body. Soaked cotton allows much less airflow than soaked wool or synthetics. Wool or wool blend socks are usually best at providing ventilation for your feet to reduce blisters and overheating. The baseball cap you mentioned before is probably also a bad idea, especially if it's cotton. A wide-brimmed hat capable of providing shade while still allowing ventilation is a much better idea. Your head generates a lot of heat, and must be allowed to properly reject it to the atmosphere in order to avoid overheating your brain.
Behavior is a much more simple category for improvements. Stay in shade, move slowly, and move even more slowly if you start to feel hot. Do not hesitate to take a break in the shade. In fact, take a break before you feel like you need to. Don't wipe sweat away unless you must. The sweat must remain on your skin to be effective. If you're producing excess sweat to where you're dripping, it may make sense for comfort reasons to wipe it away, but you wouldn't want to keep, say, a towel with you to clear your brow every few seconds.
Practice is even simpler. Exposing your body to heat while exercising will generally cause you to thermoregulate more effectively. Your body will produce more blood and vasodilate more readily, allowing the blood at the surface of your skin to "donate" larger amounts of heat to the air (if below 38 degrees) or to evaporating sweat (if between 38 and 100 degrees (I really hope you're not going anywhere over 100 degrees)). This is how you see people who live in the tropics or mediterranea go about their day comfortably without sweating like a northerner does.
Wear a wide hat of non-cotton, wear non-cotton shirt/pants/underpants/socks, move slowly, take breaks, drink water, eat salts in moderation, and keep at it.
One controversial piece of advice I'll throw out there that fits with the "practice" paragraph is if you live in a warm place, avoid air conditioning. Or, at least, use in moderation. If you want to get accustomed to 30+ degrees, make sure your AC is set to 25. Just low enough that you don't feel like you're going to die, in other words.