J3 background check conundrum
Just started J2 last week - FT contract role. J1 I have been at 4 years but it's a sinking ship. I could be let go anytime but I have been saying that for a year. J1 also highest paying.
Anyway, applied for potential J3 (as a J2) a while ago and never heard back. They fast tracked me suddenly last week and now I'm expecting an offer.
The conundrum: I messed up with J2 and they contacted J1 to verify employment. I checked do not contact, but they did anyway. HR asked and I said it's just a side gig to build savings - they were completely fine with it and even said they'd give me a recommendation if needed.
J3 lists offer is contingent on a background check. J3 has my current employer from my resume, so I run the risk of them contacting J1 to verify... again.
J3 pays less than J1 but is solid - chill boss, easy workload and decent J1 base potential for future. I'd really like to take it if possible, but am fearful of getting too greedy and HR getting suspicious if they get contacted again.
Creative ideas? Things I'm missing? Quit J1 and just roll with the 2? TIA