I said that I've never heard of Aitoff-Hammer being used to map a region smaller than the entire Earth. I should admit that I can't guarantee that, for a smaller region, the procedure for making Hammer from Azimuthal Equal Area remains equal-area.
If not, then the ordinary Azimuthal Equal Area (AEA), centered on the middle of your region, is probably the best suggestion that I know of for your purpose.
I say "probably" because, if you're mapping a quite skinny region, then there's a point at which an oblique cylindrical Equal Area map would give less maximum shape distortion. But, if only the ocean regions are important for your mapping purpose, then of course shapes aren't as important anyway.
Even if the AEA-to-Hammer process retains equal-area, an ordinary oblique Azimuthal Equal Area map might be easier, and almost just as good if your region isn't too oblong.
Because I can't find the button for starting a thread, is it ok if I add a comment on another topic here (but still about equal area maps)?:
Maps could be rated on how well they fill a rectangular space. ...on how much of their "circumscribing rectangle" they fill. In other words, since the map will be on a rectangular page or map-sheet, and because wall-space for putting up a map is typically rectangularly-bounded, it's of interest how well a map fills that rectangular space in which it fits.
The more it fills that space, the more area the map can have. That means that, on the average, the map can have better resolution, show more detail, have more space for writing, etc.
I call that attribute "space-efficiency". A rectangular map has a space-efficiency of 1.0
Circular or elliptical maps have a space-efficiency of pi/4 (.785)
The sinusoidal world map has a space-efficiency of 2/pi (.636)
Any rectangular map has a space efficiency of 1.0
Eckert III and Eckert IV's space-efficiency is about .89
The Quartic Authalic world map has a space-efficiency of 4/5, or .8
The Quartic Authalic is a pseudocylindrical world map, bounded by quartic (4th degree) power functions. It's constructed by starting with a parallel-spacing identical to that of the central meridian of the Azimuthal Equal Area, and extending the parallels out, at each latitude, to widths that make the map equal-area. That results in the outer boundaries of the map being 4th degree power functions, hence the map's name.
It's interesting that the Quartic Authalic's space-efficiency is almost the same as that of Hammer-Aitoff (Quartic Authalic's space-efficiency is just very slightly larger--.8 vs .785)
Anyway, a power function can be chosen that will give a space-efficiency about equal to that of the Eckert III and Eckert IV:
That would be achieved by a 7th or 8th degree power function. The map would be rather squarish, with rounded corners. It would have about the same space-efficiency as that of of Eckert III and Eckert IV.
Where that 7th or 8th degree map would differ from Eckert III and Eckert IV is that its top and bottom edges begin sloping equator-ward immediately with departure from the central meridian, even though slightly, at first--whereas, with Eckert III and Eckert IV, the top and bottom of the map are quite flat, over the centreal half of the map's width.
I mention that because I don't know if a 7th or 8th degree map would be a good substitute for Eckert IV (an equal-area world map).