We consider the deterministic terminating exploration in an anonymous, unoriented ring using asynchronous, oblivious, and myopic robots. By myopic, we mean that robot vision is limited within a certain distance $\phi$, computed in terms of hops. In~\cite{DLLP13}, it is shown that exploration is impossible with asynchronous settings if $\phi = 1$, {i.e.}, if the vision is limited to its own and its immediate neighboring nodes. In this paper, we study the cases where the robot visibility is limited to $2$ and $3$ neighboring nodes, respectively. In the former case, we propose two deterministic solutions in the asynchronous model for $7$ and $9$ robots. The first solution works starting from specific configurations while the other one works on any initial configuration. In the latter case, we show that no exploration is possible with less than $5$ robots. We then propose two asynchronous solutions that solve the problem using respectively $5$ and $7$ robots. The former works starting from specific configurations. The latter works for any initial configuration.