When you look up the 1870 census, do you look it up on an INTERNET site? If you do, could you let me know which site you use?
Found the census
Household Record1880 United States Census
NameRelation Marital Status Gender Race Age Birthplace Occupation Father's Birthplace Mother's Birthplace Mary ADAMSSelfWFemaleW44TNFarmerNCNC c1836 John ADAMSSonSMaleW20TNLaborerLATN c1860 Thomas ADAMSSonSMaleW17TNLaborerLATN c1863 Lee ADAMSSonSMaleW15TNLaborerLATN c1875 Henry ADAMSSonSMaleW9TNLATN c1871
Source Information: Census Place District 12, Fayette, Tennessee Family History Library Film1255254 NA Film NumberT9-1254 Page Number414D
I suggest your next step would be to get Thomas Jefferson ADAMS' (b 1862, Tennessee) birth cert for some solid info. That should give you his parents marriage details and any children born before him should be listed.
Then if need be you could get the parent's marriage certificate. These may be the only 2 certs you have to get but would be well worth it.