In many jobs you can expect a 30 - 60 day lapse period. Here are some strategies. First, you can buy continuing coverage from your former employer. This is usually very expensive and for most is not a practical consideration. But you can stay on this coverage for at least 12 months. (COBRA)
The second option is to do as Thas suggests and let it lapse, but stock up on services and supplies as late as possible to try and bridge the gap. This also may not be ideal since out of pocket and prescription carry over might not work correctly.
You can also sign up for Medicaid for short periods, I will have to let someone else speak to this because I do not have experience with it
Finally, you can choose to purchase coverage under the affordable care act. This will be tailored to your income and, will be also be expensive, but less so than other options beside Medicare.
I suggest going Medicaid, affordable care act, back end loading supplies and skipping a period of time and finally in the last case think of using COBRA.
Best of luck