$stCON$ with path of length $≥ n/2$
The NL algorithm for $s$-$t$ connectivity proceeds as follows. Start at $s$. For $n-1$ steps, guess a neighbor of the current node, and accept if that neighbor is $t$. The memory requires is two vertices (the current one and the next one), and a counter that counts the length of the current walk.

If you want to check that there is a simple $s$-$t$ path of length at least $n/2$, you need some way to ensure that in the process, vertices don't repeat. It's not so clear how to do that without storing all vertices visited so far, which cannot be done in logspace.