Just a little rough idea:
ITTL Greece won the war against Turkey in 1923, even if with lot of difficulties. By the Tittoni-Venizelos agreement, Greece exchanged a part of the land acquired in Anatolia, Antalya, with Dodecanese. Turkey managed to stabilize itself enough and to counter the other separatist movement (armenians etc...); the loss of Smyrne and Antalya is anyway a real sore spot.
In the twenties, Greece realize that a war against Turkey is inevitable. So the greeks starts a progressive fortification of the border and to modernize the army as much Greece's finance allows. Later a defensive pact is made with Italy: if a foreign power attack one of the two countries, the other will intervent. The logic behind this agreement is to avoid an alliance between Italy and Turkey: war on two fronts would be disastrous.
Ironically the greek armament rouse apprension in Turkey, which procede with an arm program of her own...
During the late 30's a series of border accidents sour the relations between the two nations. On 23 may 1939 a greek patrol, that had mistakenly strayed into turkysh territory, is attacked: the second Greek-Turkey war has started.
After some initial success, the greek army suffer a sudden defeat and is slowly thrown toward the sea. Afraid of losing the war, the greeks ask for italian intervention. Mussolini is thrilled to oblige...
At this point, I am a bit puzzled about what would happen: if the greek-italian counter offensive is successful they could beat Turkey and gain control of Anatolia; if Italy declares war on allies as per OTL, Greece would be dragged into war too.
With an axis controlled Anatolia the go-for-ME strategy becomes viable (maybe with Guderian in command of an Asian-Korp

), but what would be the allied reaction? Would they allow the whole thing to degenerate to that point? And what would be the position of Vichy Syria? IOTL they fought against the british in 1943 if I remember correctly...