Is 2n^(2/3) optimal? For instance, doesn’t there exist a constant c < 2 such that 2n^(2/3) can be replaced by c*n^(2/3)?

Let's construct a directed graph which represents the entire process. For each step, the number of coins in each town($v$) except the current position is equal to $n+d_{out}(v)$. Then we can claim that from $n+k$-th day($k\geq 0$), Hong should move to a town whose out-degree is at least $k$. Since $\displaystyle\sum_{v}d_{out}(v)=n+k$, the number of those towns are at most $\displaystyle\frac{n+k}{k}$. Hence, Hong cannot travel for more than $n+k+\binom{\frac{n+k}{k}}{2}$. Set $k\simeq n^\frac{2}{3}$, then we can derive the conclusion.