  In first-passage percolation, one places nonnegative i.i.d. random variables
(T(e)) on the edges of Z^d. A geodesic is an optimal path for the passage times
T(e). Consider a local property of the time environment. We call it a pattern.
We investigate the number of times a geodesic crosses a translate of this
pattern. When we assume that the common distribution of the passage times
satisfies a suitable moment assumption, it is shown in [Antonin Jacquet.
Geodesics in first-passage percolation cross any pattern, arXiv:2204.02021,
2023] that, apart from an event with exponentially small probability, this
number is linear in the distance between the extremities of the geodesic. This
paper completes this study by showing that this result remains true when we
consider distributions with an unbounded support without any moment assumption
or distributions with possibly infinite passage times. The techniques of proof
differ from the preceding article and rely on a notion of penalized geodesic.
