Monetary reform came to South Korea in 1966 and old hwan pieces like yours were replaced with modern won issues.
James' coin is not very valuable. It is probably one US dollar or so. The coin pictured is pretty well worn, and would have trouble fetching a dollar from a collector. If you piece is in better condition, and if a collector really needs it to complete his or her set, your coin might sell for two dollars or a little more.
If you can find one with a 4292 date (equivalent to 1959), it is more valuable. Most 4292 10 hwans are heavily circulated. An uncircluated example catalogs for $50 and up. There are many more 4294 (1961) coins in uncirculated condition. The lage Dangun dates start at the legendary founding of Korea in 2333BC.