Before you rule out carrying your current coverage with you, it might be a good idea to look into the provisions of COBRA insurance, the Comprehensive Omnibus Reconciliation Act. This is a federal law that requires insurance companies to both give you a window of coverage after employment ends and the option of picking up the coverage personally if you are willing to pay the increased cost.
Under COBRA, you are allowed to continue your current insurance plan for up to 180 days after your employment ends, and that time period can be extended another 180 days under certain circumstances. This gives you up to one full year of continued coverage with your current insurance company. The catch is that you must pay the full premiums, including any portion that was previously paid by your employer. You may have to pay a one-time administrative fee as well, but you will be allowed to keep the coverage you have while you shop around for a new company.
Additionally, many insurance companies will offer you continuing coverage based on the COBRA plan. This will allow you to keep the same insurance indefinitely as long as you pay the full premium cost. Check with your insurance company to find out whether they allow converting your current health insurance plan to a private coverage option. Doing so means that you won't have to give up the coverage you like just because you change jobs and that could be especially helpful to people with medical conditions that could prevent them from getting new coverage.
While you are using the COBRA extension, you can shop around for a new health plan that mirrors your current one. If you expect to change jobs frequently, it might be a good idea to check with an organization you belong to, such as AAA or AARP, because many such organizations offer health insurance to their members at group prices, regardless of where you work. In that way, you can choose a health insurance plan that stays with you for years, regardless of where you work, or how often you change jobs, or even whether you are employed at all.