Examples:
A careful reading of the sequence of play shows that if you use silent running for the entire turn, you are silent for all purposes before segment 1 is evaluated. If you go silent mid-turn, such as on segment 4, then you can't search that segment. The enemy doesn't consider you on SR for that segment. On segment 5, you'd be silent for the enemy trying to search for you. In addition, RX triggering is weaker since your sensors go dark before the RX activation step (as a practical matter, this is rarely relevant).
A Federation squadron enters Romulan space on segment 4 and switches to silent running. While the Federation force is penalized for being on silent running, a hypothetical Romulan force in the hex would not be forced to search if they attacked the force on segment 4. If they tried to on segment 5, they'd need to search for the hidden force first. This might happen if the Romulan force needed to use an RX before initiating combat (the segment 4 movement would light the hex up and the RX would be available on segment 5).
Added "hidden" for clarity. It was strongly implied that more than one hidden base was allowed by the text, but not stated outright.
(D7.22) You may have more than one hidden base in the hex, and they all have to be found before hex ownership will change.