I found a fix!
This has just made the game twenty times better looking! You spend so much time looking at ships at a long distance trying to ID them, now that is really a lot easier. I can't believe I didn't try this earlier.
That Super - Transparency AA checkbox is the key. You should be able to use any AA mode you want though I'd recommend Multi 4x (as I'm using above). Some small enhancement can be had with Coverage Sample but it's minor and seems noticably slower at times. As mentioned before, I don't think Supersampling is possible with anything but the highest of highest end ATI hardware, and it's probably slow there too.
nHancer is free, available at nhancer.com. First, click the Add button to create a new profile. Then add the SH4.exe to the box below called Executables with the other Add button off to the right. Now make sure that profile is highlighted and the Profile radio button is selected at the top. Select the options I've selected on the right panel. One more I have selected that you can't see is under the Optimizations tab, Texture filtering = High Quality. I'm quite sure the Super - Transparency AA is the key to it all and nothing else matters, other than making sure you have some kind of AA mode selected at the top, even Multi 2x is enough but 4x is a lot better.
Here's a different (but similar) ship, the one from the Torp Training school mission, with those settings closer and closer:

The higher resolution texture kicks in on the second to last one, but you can hardly tell, unlike the images up above.
You can even alt-tab out of the game and change this without restarting to see an immediate comparison. Here's another with virtually identical pics, from the torp training mission right at the start:
Note how this does not make any difference when you are close enough to the ship:

I can barely tell the difference, because it's past the line where the AA kicks in regardless.