In　this　paper,　some　enhanced　error　estimates　are　derived　for　the　augmented　subspace　methods　which　are　designed　for　solving　eigenvalue　problems.　For　the　first　time,　we　strictly　prove　that　the　augmented　subspace　methods　have　the　second　order　convergence　rate　between　the　two　iteration　steps,　which　is　better　than　the　existing　theoretical　results　in　Lin　and　Xie　(Math　Comp　84:71-88,　2015),　Xie　et　al.　(SIAM　J　Numer　Anal　57(6):2519-2550,　2019),　Xu　et　al.　(SIAM　J　Sci　Comput　42(5):A2655-A2677,　2020)　and　more　consistent　with　the　results　of　actual　numerical　test.　These　sharper　estimates　explicitly　depict　the　dependence　of　convergence　rate　on　the　coarse　spaces,　which　provides　new　advantages　for　the　augmented　subspace　methods.　Some　numerical　examples　are　finally　presented　to　validate　these　new　estimate　results　and　the　efficiency　of　our　algorithms.