Dear Jason,
Use 3ft of black (or a contrasting/matching color to your frame) duct tape and a folding tire under your saddle. Fold the tire in fourths or sixths. Wrap the tape tightly around the tire then around both rails and back around the tire bundle. The tape will potentially be useful for other things, and will securely hold the tire in place. Alternatively, you can tape it behind your seat tube if you've already got a (full) saddlebag, or just toss it in your handlebar/saddlebag, where it will stay dry and better protected from the elements.
However, If you're using a reasonably wide tire then I'd question the need for a spare tire on a 600K. Start with tires and tubes in good condition (i.e. almost but not quite new--perhaps 100mi on them) and carry two (or three) spare tubes, a patch kit, and material for a tire boot. I've never significantly damaged a tire wider than 23mm on a ride (other than Grand Bois Cyprèses on gravel), and even they were rideable with boots. The biggest advantage to having a spare tire is the "rainy nighttime mystery flat" problem, where you've got a Michelin wire or something difficult to find in your tire causing repeated flats. This has been a strictly theoretical problem for me, so I've not lost sleep over it.
I carried all sorts of tools, spares, etc for the first couple of years I did long rides, and started paring down things I never used.
I'm down to three spare tubes (for 700C; one for 650B), a frame-fit pump, a few Allen keys (3mm fits my shoe cleats,4mm,5mm fit the adjustment points on my bike), a tire lever, a patch kit with fresh glue, a powerbar wrapper, a spoke wrench, a fiber-fix spoke, a chain quicklink, a few zipties with a few feet of electrical tape, and a small swiss army knife with scissors, a flathead screwdriver point, tweezers, and a blade. If I'm going into remote country, I add a chain tool. I've never needed the chain repair stuff, nor a spoke wrench, but I've ridden with others who have broken their chain and/or a spoke, and both are show-stoppers. I've used the rest at one point or another.
Enjoy your first 600k. Please let us know how it went!
Best Regards,
Will
William M. deRosset
Fort Collins, CO