Quiero más cobertura en el seguro del carro, por si se daña otra llanta en la carretera.

Cobertura is the most common word for insurance coverage in many Latin American contexts (what the policy covers: towing, tire damage, roadside assistance, etc.).

• Cubrimiento exists and is used in some countries/industries, but cobertura is more widely understood.
• Más seguro would usually mean safer (or “more insurance” in a vague way), not specifically “more coverage.”

Here en is used like “within” or “under” the policy: more coverage in/under my car insurance.
You might also hear:

• más cobertura del seguro del carro (more coverage of the car insurance)
• más cobertura en mi seguro (more coverage in my policy)
  All are understandable; en emphasizes what the policy includes.

All are possible depending on region and style:

• seguro del carro = very common, conversational (many parts of Latin America)
• seguro de carro / de auto = also common; can sound a bit more generic (“car insurance” as a category)
• seguro del auto = same idea, just using auto instead of carro
  Del = de + el (literally “insurance of the car”), which Spanish commonly uses where English would use noun-noun (“car insurance”).

Por si means in case (as a precaution). It introduces a possible situation you want to be prepared for:

• Quiero más cobertura… por si… = “I want more coverage… in case…”

After por si, Spanish typically uses the indicative, not the subjunctive, because it’s treated like a practical possibility you’re preparing for:

• por si se daña (in case it gets damaged)

You can sometimes hear an imperfect subjunctive for a more hypothetical/remote possibility:

• por si se dañara = “just in case it were to get damaged” (more tentative)

Dañarse is often used for something that gets damaged (focus on the result, not who caused it).
So se daña otra llanta is like:

• “another tire gets damaged” / “another tire gets messed up”
  It avoids saying who damages it and sounds natural for accidents.

With llanta (tire), se daña usually means gets damaged (ruined, torn, etc.).
For a car or machine, se daña can also mean breaks down:

• Se dañó el carro = “The car broke down.”

Otra implies “another one (again/one more),” which suggests this isn’t the first time or that it’s an additional incident you’re anticipating:

• otra llanta = “another tire” / “one more tire”

If you said una llanta, it’s more neutral: “a tire.”

They overlap, but region matters:

• llanta = very common in Latin America for tire
• neumático = more common in Spain; also understood elsewhere
  In some places, llanta can also mean the rim, but in everyday Latin American speech it often means the whole tire.

You can, but many speakers would choose a more specific verb:

• se pincha una llanta = “a tire gets punctured”
• se poncha una llanta = “a tire goes flat” (common in Mexico and some regions)
• se desinfla una llanta = “a tire deflates”
  Se daña is broader: any kind of damage (including a blowout).

Carretera usually suggests a main road/highway (especially outside a city).
If it were a city street, you’d more likely hear:

• en la calle = on the street
• en la avenida = on the avenue