Diamond and Dybvig, in their famous model of bank runs, note that a bank regulator who commits to withdrawal freezes can forestall purely self-fulfilling runs; I no longer have to worry that my fellow depositors are going to withdraw first. A few years ago, then Fed economists Ennis and Keister noted that a regulator may not be able to commit to such a policy, desiring in the midst of a bank run to allow some agents with highest demonstrated need to have privileged access to withdrawals, and that if agents anticipate this behavior the purely self-fulfilling run can still take place. A somewhat fanciful solution is to note that, if somehow there were a liquid market for claims on bank deposits, people with high liquidity needs could sell their deposits to people with low liquidity needs; in fact, as long as the situation was expected to sort itself out within a week or two, the claims would probably trade very close to par, and, if this sort of solution seemed in the offing, there again would be very little incentive to trigger a self-fulfilling bank run.
How might one hope for such a liquid market to come into being? The best idea I can think of is to make the bank into a market maker. The bank could first, perhaps, be allowed to pay out to depositors at a rate conservatively estimated to be feasible if all depositors were paid; thus if depositors as a class are believed to be, with high probability, able to expect 25% of their deposits back in the event of failure, you allow depositors to claim deposits at 25 cents on the dollar. That puts a floor on the market. I'm hoping, though, that that would be irrelevant if the bank is also allowed to accept new deposits at a premium, and pay out on old deposits at a similar ratio. For example, suppose a 10% rule: if you come to the bank right now and give it $10, your balance at the bank will go up by $11. If the bank collects $180 this way, it is allowed to use that $180 to pay back old depositors at a 10% discount; i.e. if 30 people each line up to withdraw $9, reducing their account balance by $10, you pay out $9 each to the first 20 people in line. Once the segregated, new funds are gone, the other 10 people can choose to get $2.50 or to wait until someone comes by to deposit more money.
On as rapid a basis as is practical, you could in fact change the 10%. Again, if this is purely self-fulfilling, a 10% offer should bring in a lot of new money (which, among other things, would be somewhat costly to the bank). On the other hand, perhaps in some situations in which there is real concern for the bank's solvency, you would have people standing around waiting to pounce on any new money that came in. If the bank has a lot of new money, it could start cutting the 10% to 9%, and then 8%; on the other hand, if there are a lot of people waiting to redeem, you might start upping it to 11%, then 12%.
Again, if the bank is well-known to be solvent, this is all "off the equilibrium path", which is, of course, not to say that it's unimportant; the fact that it is credible is what keeps it from being needed. I wonder whether it or something like it would be credible under some set of institutions that exists in the real world.