Jerry,

The answer is Maybe.

Compay B would need to decide to accept your continuity log.  The majority of the time company B will do their own welder performance qualification testing just to make sure.

But your current continuity log should at least be a good tool to get you in the door to test for company B.

I think most contractors/manufacturers use the continuity log predominatly as an "internal" tool to prove that their staff is currently qualified to perform work that they bid on.

Jerry,

Company B might or might not accept your qualification log.  I agree that your log should get your foot in the door.

The good news is that your certs are still in good standing as long as you are welding.  To me, the code doesn't say that you have to be employed.  You could weld for yourself, your friends, or your neighbors.  As long as you're welding with the metals, material ranges and thicknesses and processes that your certs identify, your good.  Keep a log.  Include digital photos.

Consider.  I judge the VICA/Skills USA welding competition in my area.  It's held at a community collage.  The last few years the schools have changed all their welding equipment to one manufacture. 

The manufacture has a booth at the competition for the contestants to visit during the contest.  The manufacture loads the students up with as much filler metal and consumables as they can carry to take home.  The idea for the manufacture is to sell welders.  The idea for the students is to weld.  They might not find a job.  Keep welding.

Who knows how long a job will last? Keep welding and keep a log.  You are certified.  You might even start your own business!

Keith