Hold, meaning a grasp on something, is a very old word in the language. The OED traces it to 1300, and it is idiomatic with a number of verbs, including catch, get, lay, lose, seize, and take. We may therefore analyze I got hold of Tom's address as
of Tom's address (Prepositional Phrase Complement of hold)
(I've changed your example to show that the simple predicate is got, eliminating the distraction of need+to-infinitive.)
There are two reasons not to consider got hold as a phrasal verb. First, it's understandable from its parts. To get hold is to obtain a grasp. This is quite different from the verb in
I have to look after my brother
which doesn't mean that I have to look past him. Secondly, we can add adjuncts freely between the two words:
I got a definite though tenuous hold on the handle.
We can't generally do this with a phrasal verb:
* I have to look this afternoon after my brother.
The indefinite article may appear, as in the song lyric
You've really got a hold on me
so it might be tempting to think that the phrase started with get ahold, with ahold the (now obsolete) nautical adverb for bringing a ship into the wind to hold her steady. But the OED finds an earlier use for get hold than for ahold (the latter fittingly from The Tempest).