You seem to be asking two different questions:
- The first question appears to be: "Given a family of closed geodesics ${\mathcal F}$ which exits an end $E$, is it true that there are members of ${\mathcal F}$ which are arbitrarily far apart?"
This is true and follows directly from the definition :
A sequence of closed geodesics exits an end $E$ if for every compact $K\subset E$ all but finitely many members of the sequence are disjoint from $K$.
Now use metric neighborhoods of radius $i$ of your $\gamma$ as the compacts $K$: This yields geodesics $\gamma_i\in {\mathcal F}$ such that
$d(\gamma, \gamma_i)\ge i$.
- You seem to be also asking:
"For a closed geodesic $\gamma$ define $l_\gamma$ as the minimum distance from $\gamma$ to a geodesic in ${\mathcal F}$ (I assume, different from $\gamma$). Is it true that $\{l_\gamma: \gamma\in {\mathcal F}\}$ is unbounded?"
The answer to this question is negative. An example is given by $M$ which is an infinite cyclic cover of a manifold $N$ fibered over the circle. Let $g: M\to M$ denote the generator of the deck-group of the covering $M\to N$. Pick a geodesic closed $\gamma$ in $M$ and set ${\mathcal F}=\{\gamma_i= g^i(\gamma): i\in {\mathbb N}\}$. Then it is immediate that for $\gamma_i\in {\mathcal F}$ the number $l_{\gamma_i}$ is independent of $i$ and, hence, your set is bounded.
Of course, it is entirely possible that you meant to ask yet another question (I can think of a couple question along the lines of the above), in which case you should think carefully what your question really is.
