Post-1907 naturalization documents and WWI draft registrations are the best sources for the town name of your ancestor. If your grandfather wasn't of draft age, then look for a son or nephew born in the old country and of the correct age range.
Occasionally, Landsmanshaftn (burial societies) cover a specific enough geographic region to distinguish between two towns of similar names.
HYMAN/HEYMAN/HEIMOWITS/CHAJMOVITS: Zemplen-Dobra, Hungary > New York
KLEIN: Satoraljaujhely (Ujhely), Hungary > New York > Los Angeles
KRONOWITH: Hungary > New York
OSEAS/OSIAS/OSIASI/OZIAS: Iasi, Romania > Chicago > Milwaukee > Los Angeles
SCHECHTER/SHEKTER: Kishinev, Bessarabia > New York
SHERMAN: Iasi, Romania > New York > Los Angeles
STRUL: Iasi, Romania > Haifa, Israel
WICHMAN: Syczkowo (Bobruisk), Belarus > Milwaukee > Los Angeles