What I can recommend is that you add a batch file containing a HTTP request to cancel the last build of the job in question and then trigger the job in the next step. I think the URL below should help.
So in PostBuild step of JOB-A add a condition(single-and) step to check if the JOB-A successeded then
Execute a batch command to CANCEL JOB-B
AND
Kick off JOB-B again
Thsi should do the trick. (replace the HUDSON_URL)
HUDSON_URL/job/JOBNAME/lastBuild/stop
obviously if you may need to consider authentications if Jenkins requires a login.
Also i would say maybe in JOB-B you might want to set to block if upsteam project is building so it waits for the upstream job if its not running yet.
Hope this helped.