Perhaps, perhaps not. Depends on Italy, Turkey and Bulgaria, I think. Getting this small Empire would be dependent on British and French support - Greece was post-war a warm ally of the Little Entente, the French construction - an alliance between Romania, Yugoslavia and Greece to contain any Austrian, Hungarian or Bulgarian revanchist ideas. Britain and France were very anti-communist too.
The Megali Idea included Plovdid (Phillipopolis), Cyprys, the Dodecanese, Trabzon, and in some instances even Crimea!
The British gave the Greeks the Ionian islands 1864, I could see them giving Cyprus away for free basing rights. Peddling off some ships that needed to be scrapped to Greece (perhaps the dreadnoughts HMS Erin and HMS Agincourt, which ironically were siezed Ottoman ships) - the British get a strong and loyal ally capable of dominating the Aegean and perhaps all of the Northeastern Med. Combine this with the 120 000 exiles from Ukraine and Crimea with all of what remained of the Imperial Russian Black Sea Fleet, including the dreadnought Imperator Alexandr, and you have a pretty strong Greece.
However, the Greeks wanted Megali Hellas, not a new Roman Empire (the Byzantines always considered themselves Rhomanoi, Roman), so I don't see them going after more of Turkey. They might want to crush the Turkish state, since it is hard to defend the Anatolian coasts if someone else holds the highlands (as the Byzantines discovered), but annex? Nah.
The White Russians were none too fond of the Germans, and like German monarchists would probably despise the little corporal in Berlin.
Here's a link listing the ships Wrangel took to Constantinople late 1920.
http://en.wikipedia.org/wiki/Wrangel's_fleet