Your message has been helpful indeed! I am not at all related to the Walker family but am keenly interested in all those who were on board the General Lyon, March 29-31, 1865. I am gathering data with a view to publication - just returned from a 2800 mile car trip thru CT, NY, MD, VA, NC & SC on a General Lyon research tour.
What you say about Hart Island fits with some of the other soldiers lost on the Lyon who were also on detached duty there. The evidence shows that some of the soldiers had escorted new recruits from New York and were now returning to New York, Hart Island - presumably to do it all over again. Fort Monroe was only a way-point for the Lyon - it was bound for NY. They would have dropped some rebel prisoners at Monroe and then went on to NY to deliver the Union soldiers to QM NY.
I will look further into the Hart Island connection (which I only realized myself about three months ago) and advise if I find anything concrete. What do you have that suggests his father was directing enquiries to Hart Island?