To your specific question of claiming land in a settled metropolitan area, I really can't answer, but I highly doubt there was open land that was not owned in the city itself. It is possible a few tracts of land that were set aside for schools or general state use may have been made available at certain points.

The lands that were homesteaded/Settled of the plains and the west came from 2 main sources. One was from the Railroads, that got vast amounts of land when they laid out their railroad tracts as payment from the Federal Government. The Rail-Roads then sold lots to settlers or large blocks to speculators. The other was from the government themselves. Another source was from "land trusts" usually set up through a state government that would promote and sell tracts of land to settlers, and many also set up payments.

The Homestead Act was enacted in 1862 and lasted until 1976 and covered Federal Government owned land, which could be settled by individuals (and head of households), usually in 160 acre or quarter section plots. (the Public Land Survey System divided land one mile into squares and 160 acres was 1/4 of that)

How did the settlers find these lots and decide to go there? Mostly through advertisements and land promotions, as areas went up for sale. Family, friends that had moved to an area previously would also let people know where to go for land. The main issue for land settlement in the west was water. Having a river or a stream was where people went first. Then if water wells were successful that would be next for people to settle. The worse an area was for water, the longer it maybe un-owned or unclaimed.

After a homeowner had settled on a property under the homestead act, and had built a home structure, planted a few trees, and had to cultivate the land for 5 years it was his. They could pay a higher fee per acre to own the land after only 3 years if they chose to.

If you thought you could take on extra acreage to farm, it was probably available to you around you, as farming on the prairie and in the west on 160 acres was hard. You couldn't homestead another tract but family could or you could buy the land from another settler or a local speculator. Sometimes if the land wasn't deemed to be valuable (had no water) it could be picked up for pennies. (I've personally seen property in Western Nebraska that was traded in poker games at local army fort, and deed taken to the court house to be filed in the spring, with ownership trading over the winter a half dozen times)

The initial document that was given to the first settler was called "a patent". I've seen patents on lands next to each other that had decades between them. I've seen patents in Wyoming as late as the 1920's

Some sources to read: https://www.loc.gov/rr/program/bib/ourdocs/Homestead.html https://en.wikipedia.org/wiki/Public_Land_Survey_System

Personal source: I'm an oil and gas landman, and I trace the ownership of land and minerals from patent to present.