Is there a name for the grammatical inelegance of the original version? It reminds me of a split infinitive, but the infinitive does not seem to be involved.
It's not a split infinitive, which by definition includes the word "to" and a verb.
I'm unsure what to call the usage. It's not wrong to separate a verb and the preposition associated with it. There's nothing ungrammatic about your first example: merely stylistically clunky. In general, unless you're writing dialogue, try to keep the verb and preposition together.
That saidâ„¢, keeping verbs and prepositions together isn't always feasible. It's not illogical to say, for example, "Press together the halves while the glue dries," but "Press the halves together while the glue dries" doesn't sound stilted to American ears. John Updike needlessly "unseparated" prepositions and verbs in his old age, and ended up writing prose that sometimes read like a mediocre translation from the French.