An explicit interpolation is pretty easy to find. (I'll use the upper half plane model.)

$f$ is given by $\left( \begin{array}{cc} a & b \\\ 
c & d \end{array} \right)$.

$f$ is elliptic, parabolic, or hyperbolic if the trace has absolute value less than 2, 2, or greater than 2.

In each case you can easily find, using linear algebra, a $w$ conjugating $f$ to the usual rotation matrix, a matrix like $\left( \begin{array}{cc} t & 0 \\\ 
0 & t^{-1} \end{array} \right)$, or $\left( \begin{array}{cc} 1 & t \\\ 
0 & 1 \end{array} \right)$, respectively.

This easily gives you a path to the identity.