がけに

がけに means "on the way" or "just as you leave": something else happens, or you do something else, while you are going somewhere. It attaches to the ます-stem of a handful of movement verbs, and the common ones are fixed words: かえりがけ帰りがけに, いきがけ行きがけに, でがけ出がけに, とおり通りがけに.

The verb goes in the ます-stem, not the dictionary form, and it has to be a verb of going somewhere.

• × 帰るがけに → 帰りがけに

出がけに

出がけに is a moment, not a stretch of road: the instant you step out of the door. That is why it so often comes with something going wrong.

がけに or ついでに

ついでに is about an errand added onto another one: while you're at it. It takes many verbs and nouns. がけに only says when, on the way out or back, and only with the movement verbs above.

In everyday speech, かえり帰りに and いき行きに say the same as 帰りがけに and 行きがけに and are more common.

Before a noun

がけの describes a noun: the bus or train you took on the way.

がけ is not the かけ of たべかけ食べかけ and よみかけ読みかけ, which means "half-done".