We study a new s-process path through an isomer of ^ { 186 } Re to improve a ^ { 187 } Re- ^ { 187 } Os nucleo-cosmochronometer . The nucleus ^ { 187 } Re is produced by this new path of ^ { 185 } Re ( n , \gamma ) ^ { 186 } Re ^ { m } ( n , \gamma ) ^ { 187 } Re . We measure a ratio of neutron capture cross-sections for the ^ { 185 } Re ( n , \gamma ) ^ { 186 } Re ^ { m } and ^ { 185 } Re ( n , \gamma ) ^ { 186 } Re ^ { gs } reactions at thermal neutron energy because the ratio with the experimental uncertainty has not been reported . Using an activation method with reactor neutrons , we obtain the ratio of R _ { th } = 0.54 \pm 0.11 % . From this ratio we estimate the ratio of Maxwellian averaged cross sections in a typical s-process environment at kT = 30 keV with a help of the temperature dependence given in a statistical-model calculation because the energy dependence of the isomer/ground ratio is smaller than the absolute neutron capture cross-section . The ratio at kT =30 keV is estimated to be R _ { st } = 1.3 \pm 0.8 % . We calculate the s-process contribution from the new path in a steady-flow model . The additional abundance of ^ { 187 } Re through this path is estimated to be N _ { s } = 0.56 \pm 0.35 % relative to the abundance of ^ { 186 } Os . This additional increase of ^ { 187 } Re does not make any remarkable change in the ^ { 187 } Re- ^ { 187 } Os chronometer for an age estimate of a primitive meteorite , which has recently been found to be affected strongly by a single supernova r-process episode .