I have an umbrella policy for my LLC and a personal one as well.

However, some companies I talked with don't issue personal umbrella policy if I am not physically in US.Are there any companies who will issue umbrella policy to people in my case?

Our agents write those all the time as a stand alone policy or as a package with home, auto, and umbrella to maximize discounts.

Umbrella insurance or LLC.

A Series LLC allows you to set up multiple LLC's under one umbrella, without each LLC being exposed to any of the other LLC's legal problems that may arise.

Or should I just get an Umbrella Policy and not worry about the LLC?

If you do not start a company right away, you can make up for the lack of limited liability with a strong umbrella liability insurance policy (and I would strongly recommend that over a complicated entity structure!).

We work with State Farm because they are incredibly competitive under our umbrella policy.

But what if a self starter wanted to be a property manager under my company's umbrella because they have never done it before and would like the experience and systems we already have in place? 

They will collect the insurance premium each month in your monthly mortgage payment…As far as an “umbrella policy” that is something in addition to your regular homeowners policy (it does not replace those policies).