What you do what to ensure is that you have high coverage for umbrella-ish issues.  

Get lots of general liability and umbrella insurance covering property, run all rental business through LLC name. 2.

@wade guyThat’s why you get an umbrella policy.

I have several SF rental houses in metro Atlanta/Decatur & middle Tennessee, all with mortgages in my own name, and too many to include in my personal liability umbrella policy.  

. , Andy Krzanowsky , If these are in your personal name, I would just get an umbrella insurance policy for say $1 to $2 million.

Most times investors have their main; home,  auto, umbrella insurance with a direct writer & since most direct writers;  allstate, statefarm, usaa, etc won't write the investment properties it forces the investors to contact independents.

According to them, my primary residence homeowner's insurance liability coverage cannot be extended to cover the new property because there is a structure on the property, and an umbrella policy, apparently, will also not apply to a property with (a) no liability coverage of its own and (b) an existing structure.What are my options for getting liability coverage on this property while I fix it up?

You should strongly consider an Umbrella policy which would provide additional liability coverage in increments of $1,000,000.

Consider an umbrella policy that would kick in to cover things that your house/auto insurance doesn't.Another method is to equity strip.

I wasn't sure about the "umbrella" tax strategy where you had an S-Corp and underneath you have a bunch of LLC's with a different property in each LLC and I assumed they would go to the bank when already made the LLC and get it under it's name with no other assets and if they would get a non-recourse loan if they did that.