Never lower your rate. If you want to make yourself cheaper for the company, offer to work fewer hours, or on a fixed-term contract rather than permanent employment. For those options, you should also actually be raising your hourly rate, because you'll then have to cover more of your own costs, like taxes and insurance.

And don't let anyone squeeze you to make your actual rate lower. I once went to an interview where I was told that the expectation would be for 45 hours of billable work per week, and so I immediately told them that my salary expectation for that would be 20% higher than for a 40 hour work week. They could raise the salary, or lower the work hours, because on a per-hour basis, their pay range was suddenly no longer competitive.

They declined to negotiate that. So I eventually went to work for someone else more willing to pay me what I think I'm worth.

Don't work for free; it just encourages those who try to scam free work out of people. The only person for whom you should be doing unpaid work is yourself.

It is at times a good move, but make the offer before the rejection, possibly as being willing to work as a contractor and later go full time.