Step 1: get the paperwork that you should have been given when you left the company. That should include information on who to contact with your old employer, or the old insurance company. They also may have included information beyond what is required by federal law. This additional information could be a FAQ or a series of scenarios. They should also send additional paperwork several weeks after you leave the company.
Step 2: contact the company contact or the insurance company, and ask them about your situation. You want to know if in the case of a small medical procedure what you should do. What about if there is a large procedure? You don't need the stress of getting a insurance card rejected in an emergency room.
Step 3: understand the deadlines for signing up. One deadline is related to being able to claim continuous coverage, another may be for signing up for COBRA. Also get the exact day that coverage ended. Some employers/insurers end coverage on the last day you work, others extend it to the last day of the month. In other cases it can extend even longer if they are covering you as part of a Reduction in force.
Step 4: contact your medical provider. For many medical bills the amount of out-of-pocket the insured pays is greatly reduced by the negotiated discount and the amount covered by the insurance policy. You will have to contact the medical provider to know the size of the bill if you don't have coverage.
You will be trying to understand what you are allowed to do during this limbo period. The general guidance we give on this site, is that the retroactive nature of the COBRA decision makes it useful when the gap in coverage is short. By the time you have to decide many have already bridged the gap.
I looked for information from the federal government though they didn't discuss what to do during the limbo period:
Q18: How do I file a COBRA claim for benefits?
Health plan rules must explain how to obtain benefits and must include
written procedures for processing claims. You should submit a claim
for benefits in accordance with these rules. Claims procedures must
be described in the Summary Plan Description. Contact the plan
administrator for more information on filing a claim for benefits.
You may be faced with having to pay the whole bill now, and then if that is your only bill during the gap, then deciding if COBRA make sense for you. If you do decide to select COBRA coverage then you would file for reimbursement after paying the premium.