Leaving the unit neighbourhood on a triangular lattice

A walker sits at a vertex of the infinite triangular lattice made of unit equilateral triangles, so each vertex has $6$ neighbours. On every move the walker steps along one of the $6$ edges at its current vertex, chosen uniformly and independently of the past. What is the probability that at some point during the first $5$ moves the walker is at distance strictly greater than $1$ from where it began?

$\tfrac{95}{108}$. Compute the complement: staying within distance 1 means being at the start or at one of its 6 neighbours. From a neighbour, $1$ edge leads back, $2$ lead to other neighbours and $3$ lead outside. Tracking the probabilities of being at the start or on the ring gives a total of $\tfrac{2}{81}+\tfrac{31}{324}=\tfrac{13}{108}$ of staying inside after 5 moves, so the answer is $1-\tfrac{13}{108}$.