I checked the 1920 SOUNDEX for Joseph Hair/Hare and didn't find him.I decided to check the 1900 IT, since the two children would be on it.This is what I found:
1900 USC Indian Territory, Cherokee Nation, Welch, R28N, T21E [Welch is now in Craig Co]
Jay Haire, b May 1852, Mich, 48
Mary J, wife, b Nov 1851, Mich, 48
Robert L, son, b Jan 1881, Kans, 19
Louis S, son, b Sept 1883, IT, 16
Leota P, dau, b June 1886, IT, 13
Henry H, son, b Dec 1890, IT, 9
Emma M, dau, b Sept 1893, IT, 6
Edith L, dau, b Apr 1896, IT, 4
Claud J, son, b Apr 1899, IT, 1
You can see that Jay might be Joseph with wife Mary and two children born the years you mention.It also shows that he had been in Indian Territory for at least 16 years, since the oldest child was born here.
Welch is nowhere near Tecumseh, so you'll have to decide if this is the family you are looking for.