You're working on one end of the problem: the future landlord. But there's another end that you should be working too.
Ok, first a bit of background: If you live in the US, if you move out early, landlords can't just sit back collecting rent for the last three months (or however long is left) of your lease. We have a legal obligation to put the house back on the market and make reasonable efforts to rent it ASAP. If it takes us a month to rent it out, then you owe us rent for that month... but not for the entire term left on the lease. Of course, that's the law, then there's reality... if you move out early and the apartment is empty for a month or two, the landlord will probably keep your deposit even though they're not supposed to (assuming you left it clean and in the same condition it was when you moved in).
Here's a way to avoid that. This is what happened when one of my tenants moved out early (SEVEN MONTHS early!). He let me know he was moving out, asked me for the text and photos of my craigslist ad, put an ad up himself with his own email and phone as the contact, and showed the house to a bunch of potential tenants. He kept me updated and sent me all the info on the potential tenants (names/contact info, where they worked, when they wanted to move in). I then contacted the ones who sounded decent (i.e., wanted to move in ASAP and were employed) and had them go through my usual application process.
In other words HE did all the legwork to find me a new tenant. (If your landlord won't give you the text of his ad, you can put up your own ad, with your own photos, for free.) The place was rented before he even moved out, with the new tenant moving in on May 1; the old tenant moved out on like April 20 or something and of course he paid me the full rent for all of April.
If you do something like that, then you don't have the double-rent problem (or at least not for very long), you endear yourself to your old landlord and impress the new one, and you don't need to meet such a high bar (perfect credit or whatever) with the new landlord because you're not asking him to do you the favor of letting his apartment sit there unrented for five weeks.