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My Titleist Bag has a tether like that and the umbrella handle goes up while the tip goes in a little holder at the bottom of the bag.
Ah HA! Yes, that's definitely it. Upon closer inspection, half way down the bag is a sleeve that is clearly meant to hold an umbrella and, at the bottom, is a little ring for the tip of the umbrella. Excellent! Thanks so much for the help.
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That's a nice looking bag!
I think so, too. I got it on ebay and it just came today. I had heard so many good things about it (it's a Bennington Quiet Organizer 10), I had to pick one up. Here's a couple other shots of it:
