« first day  last day (15 days later) » 

00:22
Read the non-compete carefully, and if it really does not apply to the company you are going to, offer to sign it for a 6 or 12 month severance. Then they get to be the ones that say "No thanks." They can't keep badgering you about it if you offer to sign it. Probably be worth having a lawyer review in light of your jurisdiction.
01:21
Also, assume that some of them might contact your family members and friends. So tell your friends and family members that you're moving to a different company. That company will be your decoy. But do not directly lie to your current company. The current company is bound to find out when the new employer does an employment verification check your HR (hopefully, HR won't share the name of your new employer with your current boss, but you never know).
 
12 hours later…
12:52
If your old company is showing this kind of stalker-ish behavior, consider informing your new company about this. If they really are as petty as they seem, they might attempt to get your new company into trouble.
 
2 hours later…
15:18
@BWhite seems very, very dubious to sign anything at this point. They can promise 6 months of severance and then say "oh, you violated the non-compete, we're not paying". Just don't sign anything, go to the new job, don't bother announcing the change in linkedin for a while (say a few months). At some point the old company will stop caring.
 
7 hours later…
22:11
@Gloweye, I don't agree about informing the new company. That might be looked at as the new employee coming with baggage that can only cause headaches.

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