Bespoke is a funny thing. It is thought to mean that the customer can have whatever he specifies, when the reality is that he can have whatever his supplier feels like making or knows how to make. Take jacket linings, for example. There are fully lined coats of course, and half lined coats where the lining extends only half way down the back to promote air flow through the cloth. And then there are quarter lined jackets where besides the sleeves and shoulders there is just enough lining material to support the pockets, and self (buggy) lined versions that do away with even more of the lining material if not all of it. The latter two promote better air circulation and wear cooler than the first, but not all tailors will make them.
Not all tailors will make quarter or buggy lined jackets because to do so takes more time, moves them out of their comfort zone, or both. Less lining takes more time because one of lining's roles is to cover the seams, which take considerably more effort to make neatly. It is second nature for the Italians, with their warmer climate, but quite a bit harder to get from other tailors. And I have tried.
All that is to simply lay the groundwork for this discussion of linings, since not every style will be available everywhere. Nor will every lining material. The standard is Bemberg, a smooth inexpensive rayon, though silk is also used (for considerably more money). Silk is available in a wider variety of interesting patterns, especially if silk scarves are sewn together to make a lining, but neither silk nor Bemberg is particularly cool. Better for the heat is Ermazine, a lighter weight viscose with more ventilation, though I have had more than one tailor tell me they could not find it despite the fact that it is readily available from London's Richard James Weldon, probably the best known supplier of tailoring trimmings in the world.
Then there are the men who hold that the coolest coats are those with no lining whatsoever, though even those may have the stuff in the sleeves to make it easier to put one's arms in and out (the time required to don a completely unlined fresco jacket after airport security can cause a man to miss his flight). The challenge is that if the jacket is to have pockets it needs something to support them. When there is no lining, a double layer of the jacket's wool (or cashmere, or linen or whatever) is used as in the photograph. And two layers of wool generally speaking represent more of a barrier to air circulation than one layer of wool and a layer of Ermazine.
Self lining works fine for the interior of a coat that is lightweight but not intended to be particularly cool. It is also a reasonable choice for a heavier fresco. But for the heat, give me an Ermazine quarter lining. If I can get it at all.