Confused about the meaning of the phrase"off the top”

Even though off the top does not seem to be recorded in any major dictionary1, its meaning is roughtly right at the start, right away. Another similar idiom is right off the bat.

1Except when followed by of my head, which results in a phrase with a very different meaning.

As an example, Jimmy Kimmel said this at his Oscars monologue:

As you know, I don't have to tell anybody, the country is divided right now. I've been getting a lot of advice. People have been telling me, "It's time to bring everyone together. You need to say something to unite us." And let's just get something straight off the top, I can't do that.

                             (As quoted at Bustle and discussed on the WordReference forum.)

See also this EL&U StackExchange question: What does “off the top” mean in this context?

So Trudeau was saying that Macron takes a 40-minute press conference right away, as soon as he arrives.

From the top

There is a somewhat related phrase (take it) from the top, which is the subject of this EL&U question. For completeness, the OED explains it as follows:

Music colloquial (originally Jazz). from the top: from the beginning of a piece of music, typically for a second or subsequent time; also in extended use. Frequently in to take it from the top: to start (over) at the beginning.

Here the top presumably refers to the top of a page of sheet music (thus, the beginning of the page).

Sample usage (also from OED):
               So I guess I'd better take it from the top, and tell you everything that happened.

However, I don't claim that off the top and from the top share a common etymology. I have been unable to dig up any etymological information for off the top.