My recommendation here is that you stop thinking of this as a social situation. This is a contractual situation. The company put in 4 week notice into the contract to deal with this situation. You are effectively saying "I want to give them 4 weeks + some unknown offset" as notice. Why? If they made you sign for 3 month notice or 6 weeks notice would you be trying to give them "x months + some unknown offset" to whatever they asked for? That makes no sense. From a contractual point of view it seems you are just trying to act outside of the contract to create problems for the company where there should be no problems if you stick to the contract you signed.
Furthermore trying to operate outside of the contract by adding your unknown offset wont make your bosses life easier: he doesn't know what the offset is and you have "not really resigned yet" where the company, which has a contract with you, may not be able to afford an overlapping contract to hire a replacement, as you have not resigned.
Imagine the conversation your boss must have with any recruiters to hire for the competitive position you hold if you give him notice of your intent to resign at some unknown point. Boss says "I need a new guy to fill this role", Recruiter says "Okay, whats the start date", Boss: "I don't know, it's four week plus something unknown", Recruiter: "Er, stop wasting my time here, call me back when you have something I can work with, this is a highly competitive market, I can only deal with people who are ready to seal the deal today".
Employment has a social aspect but its unwise to apply social rules of friendship to an employment situation. This is because the company is legal entity with whom you have the relationship with and the company should always act in its own interests: not yours, nor your bosses. Treat the company with respect by dealing with it via the mechanism it asked you to deal with it: the contract you signed. Any other approach may backfire.
The only thing you could do is actually resign but say that you would consider working beyond your notice period if you have not yet got a new job and they are having trouble hiring. That assumes you don't get hired straight away. That sounds complex and unlikely to happen. Better to get another job offer in writing then resign giving your notice period.