Here is an expansion of Ian's answer. 

Dehn surgery on a hyperbolic knot or link generically gives another hyperbolic manifold.  This follows from the Dehn surgery theorem; see Theorem 5.8.2 in chapter five of [Thurston's book][1].  Furthermore, the new manifold has smaller hyperbolic volume than the original.  See Theorem 6.5.6 in chapter six.  Some further discussion and references can be found in Ian's answer [here][2].

So: fix a hyperbolic link $L$ in the three-sphere with an unknotted component $C$.  Surgering $L$ along $C$ gives an infinite family of hyperbolic links, all with volume less than that of $L$.  For explicit pictures of this, see the last few pages of chapter five of Thurston's book.  (Actually, the pictures go in the opposite direction -- he explains how the Whitehead link is the "geometric limit" of a sequence of twist knots.)

  [1]: http://library.msri.org/books/gt3m/PDF/
  [2]: https://mathoverflow.net/a/133890/1650