IANAL, but this seems questionable. The contracts I've seen mandate giving full attention to the job, so no side-jobs (but that's for "exempt" full time positions). It might get tricky with the tax-man too.
Your contract may look different than mione, but I can totally work a second job on my own time. Now it seems unlikely you could do that without impacting your current job, but the GP said they were on vacation, so you probably could in this case.
ANy non-compete would be the same if they were concurrent or sequential.
IANAL but it seems like a sticking point would be the intellectual property assignment and patent assignment agreements.
Why would the tax people care at all? They are more likely to take your salary from the two jobs, add them up, and charge taxes on the results.
Where I work, a side job is legit (with the obvious caveats, of course!)