New grad working for ~1 month now at a new job. Contacted to reapply for a position I applied 2 months ago.

Current job is a contract-to-hire position (Job A). 2nd job is a dream job w/ the government (Job B) but the process got delayed because of the shutdown. I haven't yet told them that I've been working job A but I would 100% accept the offer from job B if it came to it. So Job B is asking me to reapply because they switched over to a new hiring system. Do I include on my resume that I've started working Job A? When do I bring it up in the process if at all. I already interviewed and am being strongly considered for the position. This is for software engineering

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