Short cycling jerseys are normally designed to be ridden in summer. Unless the conditions are really extreme (e.g., North Africa in summer), one should be fine with short jerseys and shorts and sunscreen.
Use modern synthetic materials that basically all manufacturers use now, they are made to enable evaporation of sweat. Look at what people who are paid to ride their bikes wear. They only put on long sleeves when it is colder or when using skin-suites for time trials. But they do not ride in extreme conditions in summer (southern France or Spain in summer is not extreme yet).
What they do in very hot stages is to pour water on their bodies or even to put packs of ice into their jerseys. Unfortunately, water is a sparse and heavy resource so we usually cannot waste it, but if you are at a water source, do get a wash, do wet your headscarf or a cycling cap in water, and so on. The evaporating water will cool you as nothing else could.