Below are maps of Gangwon Province (江原道) and Samcheok Dohobu (三陟都護府), which includes a map of of Ulleungdo (鬱陵島). Off the east shore of Ulleungdo is an island labeled as "Usan" (于山), which was almost certainly Ulleungdo's neighboring island of Jukdo (竹島). Jukdo is about 2.2 kilometers off the east shore of Ulleungdo. The maps come from the Korean atlas pictured on the left, which is stored in the Koryo University Library.
The Koryo University Web site says the atlas was made sometime between 1835 and 1894, but I think it can easily be dated between 1884 and 1894.
If you look at the Gangwon Province map below, you will see a list of Gangwon Province official titles on the map. Among the officials listed was the "Ulleungdo-jang" (鬱陵島長 - Ulleungdo Chief), which I have circled in red. Next to it was written "Pyeonghae, Samcheok-po Jin" (平海三三陟浦鎭). This is the clue to the date of the map since the position of "Ulleungdo-jang" was not established until 1883. Moveover, we may be able to pinpoint the date even closer based on the fact that "Pyonghae" (平海) and "Samcheok-po Jin" (三陟浦鎭) were written next to "Ulleungdo-Jang," which suggests that the Ulleungdo-Jang came from Samcheok-po Jin.
In 1884, the duties of Ulleungdo-Jang was transferred from low ranking officials dispatched to Ulleungdo to the Samcheok Commander (三陟營將) and then to the "Pyeonghae County Head" (平海郡守). However, during this time the title of "Ulleungdo-Jang" (鬱陵島長) was changed to "Ulleungdo-Cheomsa" (鬱陵島僉使). The title of Ulleungdo-Jang (鬱陵島長) was not reestablished until 1888, when the duties were handed over to the "Manho" (萬戶) at Weolsong-jin (越松鎭), which was in Pyeonghae County (平海郡). The position of "Ulleungdo-Jang" (鬱陵島長) continued until 1895, when it was upgraded to "Ulleungdo-Gam" (鬱陵島監 - Island Supervisor). That means the map would have had to been made sometime between 1884 and 1895.
Based on the fact that "Samcheok-po Jin" (三陟浦鎭), instead of "Weolsong-jin" (越松鎭), was written next to "Ulleungdo-Jang" (鬱陵島長), and on the fact that the map appears with a map of Samcheok Dohobu (三陟都護府), the map seems to have been made between 1884 and 1888. Even though the official title during that time period was "Ulleungdo-Cheomsa" (鬱陵島僉使), "Ulleungdo-Jang" (鬱陵島長) could have still been used as an unofficial title, or the mapmaker could have just made a mistake and used the old title. At any rate, the map was made after 1884.
The significance of this map is that it shows Ulleungdo with a neighboring island labeled as Usan (于山) rather than the Jukdo (竹島) that Lee Gyu-won found during his inspection in 1882. That suggests that inspectors discovered that the neighboring island was being called both Jukdo and Usando. It also means that this was the last Korean map of Ulleungdo, which is still known to still exist, to have been made before Korea's 1900 Imperial Edict 41, which made the main island of Ulleungdo (鬱陵島) and its neighboring islands of Jukdo (竹島) and Seokdo (石島) a county of Gangwon Province.