Rowan -
Your way of camping is the best approach to random camping in the Great Plains and Canadian Prairies - where there is not a lot of public land. The public lands (after they were seized from native peoples) of both countries were designed as a huge grid system with each section, ideally, 1 mile square. Of course, on a spherical surface something has to give - and there are always some sections that are odd shaped or off center. In the north-south direction, every 24 miles there is a "Correction Line" where the sections do not line up exactly. When farm roads run along these line, they always have a jog. These are great areas to camp since there is usually an extra triangle of land in the right of way.
Such is the case in Greenwood County, Kansas - on the TransAm Trail. If you look due east of Eureka you can spot the jogs. But there are tons of other potential places. If you look at the paved road running southeast from Eureka - there is a dirt road 1 mile to the west of Hwy 99 that is closed just past the bridge. Nice and quiet - right on the riverbank.
J