There's a couple of nuances to bear in mind. If your employer is subject to COBRA requirements, as Ben Miller mentions, you'll be able to continue your coverage via COBRA as long as there was no gross misconduct.
- Continuation coverage must be continuous, you can't have a break in coverage. (e.g. if you lose coverage June 30, you cannot elect to have COBRA begin 8/1)
- You'll pay the full gross premium, not just your contribution, plus an up to 2% administrative fee. (It wouldn't surprise me if your monthly COBRA premium is several hundred dollars)
- If your current employer offers multiple medical plan options you will only be offered continuation under your currently elected plan option.
- You can continue your coverage via COBRA even if you are eligible for a new employer's plan. But, if you waive your new employer's coverage when it's offered to you, you will not be able to enroll in your new employer's plan until the next open enrollment or your next qualifying event.
You have 63 days from your coverage loss date or your date of COBRA notification, whichever is later, to actually elect COBRA continuation. It's pretty common to wait until you actually need the coverage to make the election. Though don't wait until too close to the deadline.
Understand the COBRA offering before you make your decision. The price is likely to carry some sticker shock. If possible, understand your new employer's plan before making your decision. It may actually be advantageous to you to walk away from the deductible you've paid to take advantage of your new employer's plan. If you're on an HDHP with an HSA account, it may also be in your best interest to load up that HSA account with tax-free money while you're still enrolled in the HDHP. You can spend HSA funds on qualified expenses but you can only contribute to it while you're enrolled in a qualified HDHP.
Depending on your age it may even be financially advantageous to buy yourself a $0 deductible gold or platinum individual plan as you transition from one employer to the next.