Is $L \subset 1NL$ when $L \neq NL$?
Define the language $BACKPOINTER$ to have words of length $n+t\log n$, divided to $1+t$ parts, one of length $n$ and the rest of length $\log n$, by commas such that $BACKPOINTER=\{(x,p_1,\ldots,p_t)\mid x_{p_i}=1 \forall i\}$.
It should follow from some standard one-way communication complexity bound that $BACKPOINTER$ needs at least $t$ bits of memory stored after $x$ is read. So for any $t\gg \log n$, $BACKPOINTER\in L\setminus 1NL$.

ps. Note that $BACKPOINTER\in 1co\text{-}NL$.