Use of “Después” in this sentence

[Rusty]

Here 'después' can be translated as 'then', and it wouldn't be unusual to see it written «para después lanzarse», if that would have made more sense to you. But it is OK as written.

It could also be translated as 'later' or 'afterwards', in which case you could mentally substitute 'luego', if that helps any.

I would even venture a translation of 'in order to'.

Here are a few translations:
Around them, other seagulls ...
... swam or flew and then dived into the water in search of small fish.
... were swimming or flying, then diving into the water in search of small fish.
... were swimming or flying and later launching into the water in search of small fish.
... were swimming or flying in order to shoot into the water in search of small fish.