Given that the technology of transportation and moving supplies to the city didn't change much from The Wheel in the Bronze Age to the mid-18th century, any extension of city radii in that period (Antiquity and Exploration Ages) would be based strictly on Political Control. After that, the advent of more modern road making and especially railroads and steam power enormously increased the size of the city radius, and that could be modeled in the game by a Tech-based increase. Either Modern Age Tier 1 Steam Engine or Tier 2 Urbanization Techs would work, to extend the overall city radius to, say, 4 tiles in all directions.
Earlier, a general extension over land tiles just isn't technologically possible. Wagons or carts or pack animals over land, regardless of how good your roads were, couldn't haul anything more than about 100 - 150 km before the draft animals had eaten the equivalent of everything they were hauling by weight and you wound up with a Zero Sum adventure.
The way around that was to use water - river, coast, etc - transportation, which was an order of magnitude more efficient. I suggest that the Game Start 3 tile radius could be extended 1 extra tile on a river tile, or 1 extra tile on a navigable river and on either side of the navigable river. On the coast, you could extend an extra tile along the coast. Placed along any kind of river, then, that could give your Antiquity/Exploration city a bunch of extra tiles to work, and depending on the configuration, at least 2 tiles along the coast.
There is, after all, a very good reason why every major pre-modern city was built with access to a river or the coast or both.
There, when your city grows, you get a "claim", which can be used to take a region. Regions within a radius of 3 costs 1 claim, beyond that they cost 2, so you have to save up for it. 
