hi amelia,
I am not going for honeymoon after AD, me and my hb already decided to go to Greece and Italy for HM, so definitely have to travel during near summer time for the Greek islands. We choose to go during end May early June period cos' it's not yet the peak season, it's call mid season, so not so crowded.
We have not booked any tickets yet, but I have more or less know which airline we are going to take and which hotels we want to stay in. The trip will be F&E, we prefer this way so that we can plan our own itinerary and travel at our own pace.
If you really want to go HM straight after AD, like what Cheryl mentioned, can consider beach HM, places like Guam and Saipan, or Caribbean cruises. Me and my hubby will be in Miami and Caribbean end Feb, not for HM but it's a biz conference + personal tour trip.
Happy searching for your HM destinations!