Since Asparagus likes to grow in soils that are good and healthy and drain well I would look at what I have and add compost or other forms of organic matter rather then more soil. With the very well draining sand I have a good mulch over the winter seems to help hold some moisture the next growing season that would otherwise drain away before any plants could utilize it, but I might not do that if I had a not too well draining clay soil.

"Also I was wondering if anyone covers the bed with a mulch that will decay under the snow and feed the soil when it warms in the spring?"

I cover the bed with straight horse manure after the stems brown, preferably removing them first.

In the spring when they start to break through I dust with corn gluten meal.

Thanks for the helpful ideas, I think I know exactly what to do.
It is the best bed I've ever done and I didn't want to lose it. :)

I mulch with whatever I have even though snow is a rarity here. Also, every other year or so, I salt it with rock salt (never table salt). Asparagus also does well with a clover livingmulch.

I would think that the root mass of the clover would eventually create problems for the stems coming up from underneath. This doesn't reduce the number of stems over time?

Maplerbirch, I share your thinking there. But, if the 'gus is in a rich environ, deep planted as per usual, there could be a coexistence.
A good 'gus patch can yield for twenty years but not with competition, including female starts. Seems the clover would become that.

We do have a lot of roadside 'gus, that is under a tangle of quackgrass roots, so maybe?
Then I wonder if the roadside stuff would do better without the quack competition.
The clover is an interestting thought. Thanks for the idea.

Wow some really helpful solutions here. As long as u have ample sun and just the right amount of water - you should be fine :)

Here is a link that might be useful: Organic Direct

Never had a problem with the clover--I like to use one of the pink clovers cuz it not only adds nitrogen, but sure does look pretty. Any of the clovers will work, but you may want to use subterranean clover if you're worried about the asparagus coming through. It dies back making mulch, then reseeds, etc. I seeded pink clover in my asparagus yesterday.