I've never seen/experienced the '24 after' restriction before but I have had a couple other scenarios with DH seat selection. It depends on what 'class' ticket is used (M-class, Y-class, etc.), and how full the flight is when the DH is booked.

I've had a mid-rotation DH as a line holder (trip built previous month) where any available coach seat (including Econ Comfort) was selectable when I added my name to the record locator. I've also had DHs on reserve-call-out rotations where the flight was overbooked/full and it wouldn't allow me to change from my assigned, middle-of-three, steerage class seat.

In most cases, when you check in (24hrs), regardless of which 'class' ticket they used, if there is a E.C. or exit-row seat available, you should be able to select it. You can also 'request upgrade' at the same time; but don't count on the first class on any domestic flights though -- Medallions win out over employees.