Not sure if you’d be willing or able to pull it off, but might consider offering X number of months in advance.
With our current LL (since mid-2004), we paid 12 months in advance for the first year and 6 months in advance from that point forward.
Of course, you really need to trust the LL & I’d suggest having an atty come up with an agreement that YOU give the LL regarding escrow accts, LL credit reports, etc.
With a good FICO, advance rent payments & good references, you will have no problem negotiating whatever you want. Also, if you plan on renting for awhile, you could consider offering a two or three-year lease agreement. Talk to an atty & see what you can come up with.