Fixing soil salinity can be very difficult. If you've got fairly good drainage, adding gypsum and flushing should help, but it may take time. I think you need to add the water all at once for it to be effective because you're really trying to flush the salt out, not just dilute it. If you water, but not enough to flush, you could be adding more salt (drinking and irrigation water has small amounts of salt so irrigating at normal levels will slowly build up salt).

The bed is small enough that you could excavate the soil out to a depth of two feet and come up with just a bit over one cubic yard of material. That isn't very much; of course, it depends on how much you like to dig.

It isn't necessary to dispose of the dirt if you do not want. Reuse most of it in a 50:50 ratio with finished compost to buffer the salt.

Terran

Gypsum will aid in cleaning the salt buildup from that soil. Gypsum is used in arid areas of the country (sodic soils) for that reason, although there are different kinds of salts to contend with there. Flooding the area, several times, could wash that sodium chloride out of the soil, eventually but the gypsum would be faster.

De-icing salt is often calcium chloride. Will gypsum also work for this?

I've not seen any research that says it will but my experience does indicate applying gypsum even where calcium chloride was used helped.

I read this last night on pages 155, 156 in Our Future Reborn by Dr. Teruo Higa.

"Another benefit Egypt hoped to gain from EM (Effective Microorganisms) was a remedy to soil salinity, which had dealt a fatal blow to desert agriculture. ... Amazingly, in some places where EM has been used, soil previously damaged by salinity has been restored and the salt crusts that accumulated on the surface have vanished along with the large white spots where salt permeated the soil."

Terran

P.S.

If you decide to use EM-1 it can be what they call "activated" to make 22 times the amount purchased. It is easy to do; I've done it.

"EM-1® Microbial Inoculant products can be used directly from the bottle or they are grown one time for economical purposes. This "growing" process is called Activation (or extended). The normal activation process is to take one part EM-1® and make 22 times the amount through a simple fermentation process, similar to making beer or wine. ... (T)his simple process normally takes anywhere from 5-10 days to complete when done properly."

http://www.emamerica.com/effective-microorganism-
products/activated-effective-microorganisms

T.

Here is a link that might be useful: EM1® Multi-use Size