Tricky.
The insurance term that you are looking for is called "Missed Connection". See
here or here. It's not a particularly well advertised feature, so you will have to dive into the fine print.
I did look up a few policies and some do indeed include coverage for missed connections. It's typically limited to a certain amount ($500 seems common) and often restricted to "tours and cruises". What I saw was certainly NOT cheap, but you should get a quote with your specific details.
As typically with travel insurance, "missed connection" coverage comes with a lot of exclusions, including "self connections", and the policy documents are often dozens of pages long written in hard to understand "weasel language". It's a good idea to read it carefully before shelling out.
People have told me several times that I should just buy "insurance"
That feels a bit of a knee jerk reaction and is frequently not based on careful research and thorough risk assessment. The simple fact that every travel provider tries to sell you insurance at check out is a clear indicator that it's great deal for them and a bad deal for you.
I'm a frequent traveler and I never use travel insurance. It makes no sense to me financially. Travel insurance that actually has reasonable coverage is fairly expensive and the few cases where I had to deal with an issue would either not be covered anyway or didn't warrant buying insurance for all trips that I do.
layovers from different airlines,
Sorry, but unless you are a fairly experienced traveler who can do a reasonable risk assessment and is comfortable with some risk and making alternative plans, you should NOT be doing this. If you want to go from A to C, buy a ticket that gets from A to C. Yes, that can be more expensive, but it will almost always be less expensive than a self connection going sideways and it buys you "peace of mind". At the very least make sure you have at least 4-5 hours between flights.
CAVEAT:
You should NEVER be without insurance for medical emergencies.
That certainly applies to travel as well. The musings above are primarily about trip cost insurance (cancellation, delay, lost baggage, interruptions, missed connections, etc).
Medical insurances is completely orthogonal topic. If you already have primary health insurance and it doesn't cover travel you should definitely look into "medical only" travel insurance. Prices and coverage vary wildly depending on where you are from, where you are going and what your primary health insurance is.
Personally, my primary health insurance coverage makes no distinction between home or travel, so medical travel insurance makes no sense to me. But this is different from case to case.
