Welcome to the forum, hodag! What parts are you from? Any pics of your bike?
What's great about the KLR is that you can use it for many purposes. As seen in the 1st pic, a lot of the more off-road worthy tires are still good on the pavement for even longer rides.
You can even do some exploring with it loaded down, if you so desire:
My longest trip was a trip from Philly to NC. This is the tire I ran:
It is the IRC 110. The company claims it to be a 90/10 street tire, but I think they underestimate it; I say it is at least a 70/30 if not 60/40, but still very good on road. Got 9k miles on the front and still had plenty of miles left when I took it off. Rear got nearly 4k, but I've always been rough on tires. Price is very fair, especially compared to others in it's class. When I got this tire for the 1st time, it was because it favored the very expensive Metzler Enduro 3 below:
Our selection of tires back in the mid-90's were very sparse and I couldn't afford the very popular Metzeler, so on went the IRC's for less than half the price as the Enduro 3.
If the tires you have on your bike now are more knobby than these, then maybe going to these 70/30 tires would be the ticket. I know you mostly do pavement now, but what's good about the 70/30 is that it doesn't normally give up much on the pavement compared to the 90/10, but definitly is a more friendly tire when you do hit the FS roads and other dirt.
Whatever you use your KLR for, I'm sure you will have a great time.....