Consider a reversible random walk on (say) $\mathbb{Z}$. Are there any estimates for the probability $\mathbb{P}(\tau_n=m<\tau_0^+)$ where $\tau_n$ is the first hitting time at site n and $\tau_0^+$ is the first return time to the starting point 0?

The conductance decreases exponentially in distance, but the Simple random walk case can be already helpful for me.