The KLR 650 C model was sold in Europe up until the late naughties (correct me if I'm wrong) The A model is still on sale in the USA/Australasia and apparently Ecuador. Actually the C model was in many ways a better bike than the A, as it had stronger suspension and better brakes. If you can find a used one of those, they'd make a fine overlander but you'll definitely have to change the tank.
I've seen some French dudes give the KLR C model the supermotard treatment and it looks quite cool.
For my 2 rupees, I've owned a A model for the last 10 years and it has rarely let me down in 30,000 miles of overlanding torture. I've pulled 2 XR's out of the Sahara and seen off burning KTM 990's and snapping BMW frames.
Mind you it needs some prepping before serious overlanding can be considered.
I did try to buy parts for it in the UK/Europe and all I could come up with is WeMoto for some spares. Different story in the States.
Join the brotherhood, become a KLR-ista! Orale!