If you’re willing to sign the offer, you have manufactured leverage you can use to squeeze out additional compensation with high probability. Tell the recruiter you’re willing to sign the offer immediately if they can give you X, where X is a bit higher than the current offer.
A good choice is to ask for an extra $10K-$20K in sign-on bonus. Companies are willing to do this since it’s a one-time cost but they can guarantee you sign. But $10K is be life-changing for many people.
Making the final request focused on salary or equity is a bit more difficult since it often requires more people’s approval. Your goal in this final round of negotiation is to propose something quick and a no-brainer for the company.
Phrasing:
I'm eager to use my experience to have impact quickly at [COMPANY]. I'm willing to sign the offer today if you can add a signing bonus of $15,000. This would make the offer compelling for me so I can commit to [COMPANY] for the next phase of my career.
Ideally you want things in writing. Some recruiters will tell you they can’t put it in the formal offer until you’ve verbally agreed, which is fine, but in that case, you can still ask them for a quick email recap with the numbers included.