Sorry Cam, I know that the question was meant for Sam but I want to put in my two pence.You see a ship, just faintly, by your estimates, maybe 10 miles away.
They are not responding to the distress rockets
Would you send your passengers to row over to that vessel and have her come at once to your ship?
Your question can be considered either "generally" or as specifically relevant to the Titanic's actual situation.
Let us take the Titanic's actual situation first. If the other ship by the Captain Smith's estimate was about 10 miles away, it would have taken a lifeboat over 3 hours just to reach it, let alone anything else. By then the Titanic would long have been on the ocean floor. Smith more or less knew that by what Thomas Andrews had told him after a proper damage assessment and so under those circumstances Smith's order made no sense.
On the other hand, if Andrews' calculations had shown that the Titanic would stay afloat for 8 to 10 hours, then Smith's order would have made perfect sense. Also, by then other ships could have also come for the rescue.