Do not quit your current job to search for a new job.
Like it or not, being currently employed is a very strong signal in favor of you being employable. If you quit, prospective employers can't tell if you've been fired, or if you're the type of person who can't keep a job, or any other number of negative signals. (Yes, I know it's not perfect or fair. Please don't shoot the messenger).
> Maybe it's possible to do more interviews and get offers to compete against each other?
Possible, but not as helpful as you'd think unless you're a top-1% candidate with a resume to back it up.
Focus on finding a company you want to work for, with people you want to work with. If you ignore everything and focus only on money, one of two things will happen:
If compensation is your highest priority, just skip straight to interviewing at FAANG companies. It's extremely rare to find small companies willing to pay FAANG level compensation.
If FAANG is not an option, be careful about high-paying companies that aren't well-known. Some of them are good places to work, but others lure in inexperienced engineers with high salaries and then work them 80hrs per week until burnout, then replace them with the next round of engineers.