In　order　to　reach　the　best　numerical　properties　with　the　numerical　manifold　method　(NMM),　uniform　finite　element　meshes　are　always　favorite　while　constructing　mathematical　covers,　where　all　the　elements　are　congruent.　In　the　presence　of　steep　gradients　or　strong　singularities,　in　principle,　the　locally-defined　special　functions　can　be　added　into　the　NMM　space　by　means　of　the　partition　of　unity,　but　they　are　not　available　to　those　complex　problems　with　heterogeneity　or　nonlinearity,　necessitating　local　refinement　on　uniform　meshes.　This　is　believed　to　be　one　of　the　most　important　open　issues　in　NMM.　In　this　study　multilayer　covers　are　proposed　to　solve　this　issue.　In　addition　to　the　first　layer　cover　which　is　the　global　cover　and　covers　the　whole　problem　domain,　the　second　and　higher　layer　covers　with　smaller　elements,　called　local　covers,　are　used　to　cover　those　local　regions　with　steep　gradients　or　strong　singularities.　The　global　cover　and　the　local　covers　have　their　own　partition　of　unity,　and　they　all　participate　in　the　approximation　to　the　solution.　Being　advantageous　over　the　existing　procedures,　the　proposed　approach　is　easy　to　deal　with　any　arbitrary-layer　hanging　nodes　with　no　need　to　construct　super-elements　with　variable　number　of　edge　nodes　or　introduce　the　Lagrange　multipliers　to　enforce　the　continuity　between　small　and　big　elements.　With　no　limitation　to　cover　layers,　meanwhile,　the　creation　of　an　even　error　distribution　over　the　whole　problem　domain　is　significantly　facilitated.　Some　typical　examples　with　steep　gradients　or　strong　singularities　are　analyzed　to　demonstrate　the　capacity　of　the　proposed　approach.