According to Esquire's Encyclopedia of 20th Century Men's Fashions, a gray jacket and white trousers like the ones worn by the man on the right in the also-from-Esquire illustration were once the most popular warm weather casual clothing combination. I have written before that it is a look that deserves resurrection, particularly combined with white bucks and a light-weight scarf worn over a tee shirt or polo. And, given a competent tailor, it would not be hard to do so.
The jacket is double breasted, a style that is counter-intuitive for warm weather, but the patch pockets hint at a quarter or buggy lining to permit air flow. The little additional warmth from the extra material across the chest is a small price to pay for fashion, particularly if it is executed in a high twist cloth like Fresco (J&J Minnis has a suitable light gray). And the combination of those same pockets and a set of mother of pearl buttons would ensure that no-one is likely to confuse it with an orphaned suit coat.
Cream Irish linen is probably the best bet for the trousers. Dormeuil had a white 10 ounce flannel but I understand it is gone now, and flannel does not wear all that cool to begin with. For the rest, white buck is difficult but reversed calf shoes are not hard to find while Drake's London makes modal scarves that are light as air.
And that is one way to resurrect a great look from the past.