I think you're golden as is. 12 inches is IMO quite a sufficient depth for plenty of stolons to develop above the potato seed. I manage quite less than that in fairly heavy soil and get quite a crop of taters. I have seen many examples of "potato towers" attempting to have 3 feet or more of buried stem, with spectacularly poor results. Just let the leaf canopy do its job now and you should have a good crop in that fluffy raised bed.

agreed with tcstoehr - it seems fine without covering the potato stems with more hay. i have done both ways and i only ever get potatoes growing below the seed and never any on the area i have hilled in so i stopped hilling/ with no straw cover. I plant them in small mounds straight into tilled soil and just leave them to grow without bothering it. this years plants look huge and mine are flowering nicely today. i found spiders really like to live in my hay piles, so no hay, and the bare soil between the potato mounts is getting covered by Suyo Long cucumber vines. Not sure if they are noted companions or not but i'm liking the combo.

Thanks for the responses - thats what I was hoping. If I pile up straw (which is hard to find around here) I'll be over the edges of the bed pretty fast and I think it will just spill off into the paths.

I am amazed at how potatoes grow so much faster than anything else in the garden!

While it may seem that all the information available about growing potatoes is doing so in straw there is still a lot of good information available about growing them in soil.

Here is a link that might be useful: Growing potatoes in soil