This is more of a comment intended to provide examples. One can construct a reasonably large class of example of metrics on $T^2$ satisfying your property. I believe moreover that this class of metrics is stable with respect to small $C^{\infty}$ perturbation.
An infinite geodesics on surfaces without self intersections a closely connected to geodesic lamintations. Now, hyperbolic surfaces have plenty of geodesic laminations. So we can do the following. First take a hyperbolic metric on $T^2$ with a cusp and then smoothen the cusp close to infinity. Then there still will be a geodesic lamination in the hyperbolic part of $T^2$. For a concrete example, look at Fig 11 here: (Geodesic lamination on surfaces by Bonahon)
https://www-bcf.usc.edu/~fbonahon/Research/Preprints/StonyBrookProc.ps
Or one can use a different construction, identifying a part of $T^2$ with a part of hyperbolic surface of genus $2$ as on Fig 5 in the same text. In this case one will get in infinite geodesic in $T^2$ that accumulates to two closed geodesics on both ends.
I believe that both examples are stable under small perturbation of the metric.