Is it ok to start looking for other postdoc positions?
Actually, it sounds like it is essential for you to to look for alternatives. It is improper for the PI to hold you in jeopardy and he should be more clear.
But your career is at risk if you just &quot;hope for the best&quot;. You also need to &quot;plan for the worst&quot;. It is more than OK.
Whether you let him know or not is a matter of your judgement of his personality. But it might be fine to say (a) I'd love to stay, but (b) I recognize it may not be possible and need a backup.