I am concerned as you mentioned if someone is bit by dog, trip and fall, slips on the steps... etc.
There are so many layers to this. Insurance is only one of them. If you are a high net worth individual, or live in California, or the property is in California, or all, the risks are real and they are everywhere. You cannot eliminate those risks. But the best attorney I have ever used always talked about making the hill as steep as possible for a claimant or litigant to have to go up.
Suppose your unattended car in the garage gets stolen for a bank robbery. Shots are fired and people die. You will be sued, no question. I believe your vehicle coverage is for motor vehicle type operations so it won't provide coverage nor cover costs for defense. Big risk. Your rental or landlord policy probably won't cover this as it was your personal vehicle.
Suppose stairs collapse and someone is paralyzed. Or an old ungrounded electrical plug shocks someone. (You did disclose that and have your tenant acknowledge it in writing-- I hope?) You "should have known" to fix such-and-such and you must have "failed" to properly inspect the property prior to tenant move-in. If you have the deep pockets, it's always your fault.
If the rented real property is in your personal name, then the many other assets, properties, bank accounts, etc. in your personal name are opened up to a claimant or litigant. A successful claim could wipe out all the personal assets. Those claims could be from a dog bite, lead paint, or a meteor that falls out of the sky and hits your tenant. The basic rule is that if anything bad happens, you are the wealthy greedy property owner that has more than your fair share, and you will get sued. Sorry for such dark commentary but I believe it is legitimate opinion.
I can tell you how I have dealt with this:
1. Put legal ownership of rental properties in a LLC or corporate entity, not held personally. This builds something of a wall around your personal assets. It is easy to move the properties in/out of LLC vs personal ownership if that is something you occasionally need to do. The one time I have done this the title company took care of it for free.
2. Purchase landlord insurance for the properties in the name of the LLC. Require tenants to maintain renter's insurance.
3. Keep good separation between the LLC and your personal bank accounts and assets (no commingling.) Ideally, don't manage the rentals yourself to minimize claims of discrimination, or claims that arise from the many things you might *fail* to do. (I do manage my rentals myself, however.)
4. Don't commingle personal property (left at the rental site) with your rental. Any personal property left at the real property being rented could be a tripwire to lawsuit/exposure. In my opinion it is a big gaping potential liability.
5. Purchase personal insurance for personal assets.
6. Purchase umbrella coverage that sits atop all your assets (personal, vehicles, LLC.)
I realize you are probably storing personal items because it is very convenient and seemingly cost-free to do so. That could get negated quickly with a claim. Another possible source of a claim is inadequate supervision of your tenant. My leases allow me to inspect the property every time the clocks change for cleanliness, safety hazards, etc.
The more you can make the hill steeper, and the more you can reduce the potential size of the economic bonanza if a claimant is successful, the more likely you are to never be sued, and never use the insurance you purchase. Sorry to digress but there are so many tangents to this insurance issue ....