Let $\Sigma$ be a compact surface of genus at least $1$ with one boundary component, equipped with a hyperbolic metric so that the boundary is geodesic. There is a version of the collar lemma that says there is a collar neighbourhood $C$ of the boundary such that no simple closed geodesic on $\Sigma$ enters $C$.

Given a curve, I am interested in measuring what length of it lies outside $C$. More precisely, I am curious about the following: does there exist a constant $d>0$ such that, if $\gamma$ is an essential simple closed curve (not nullhomotopic and not homotopic to the boundary component), then $$\ell(\gamma|_{\Sigma\backslash C})+d \geq \ell(\tilde{\gamma})$$ Here, $\ell$ is the hyperbolic length, $\tilde{\gamma}$ is the unique simple closed geodesic homotopic to $\gamma$, and, to be clear, $\ell(\gamma|_{\Sigma\backslash C})$ refers to the length of the restriction of $\gamma$ to $\Sigma\backslash C$ (this is a union of disconnected curves). Here $d$ should depend on the hyperbolic metric and the choice of $C$, but not the homotopy class of the curve.

The reason I add the "$+d$" is that one could cook up the following example: take a curve that enters $C$ once, and then replace the region that enters $C$ with a straight line connecting the boundary points at which the curve crosses. We then get a closed curve homotopic to $\tilde{\gamma}$, and the length is at most $\ell(\gamma|_{\Sigma\backslash C})+d$, where $d=\ell(\partial C)$.

It seems true for just about any example I draw, but any proof I attempt to write down becomes terribly buggy and complicated.