The Euler Sombor index of a graph is the sum, over its edges \( uv \), of the quantity \( \sqrt{d(u)^2+d(v)^2+d(u)d(v)} \), where \( d \) denotes the vertex degree. Khanra and Das [MATCH Commun. Math. Comput. Chem. 94 (2025) 525-548] classified the trees with the first five minimum Euler Sombor indices. Parts (c) and (d) of their Theorem 3 are incorrect: the family presented as fifth-minimal is in fact fourth-minimal for every \( n\ge 7 \), and the value printed for the fourth minimum, \( 4\sqrt7+4\sqrt{19}-18\sqrt3+2\sqrt3\,n \), is not the Euler Sombor index of any tree. The error is a miscounted edge contribution in the proof, where the edge joining two adjacent vertices of degree three is given the weight \( 2\sqrt3 \) of a \( (2,2) \)-edge instead of \( \sqrt{27} \). We give the corrected classification of the first five minima for all \( n\ge7 \) and, using a decomposition of the index over the threads of a tree that reduces the problem to a finite comparison of skeleton penalties, we extend it: for every \( n\ge11 \) the fifth through thirteenth minimum Euler Sombor indices are attained exactly by the nine families of subdivided double stars \( S(3,3) \) determined by the number of subdivided leaf threads and the subdivision status of the central edge, in an explicit order governed by the quantities \( \delta=\sqrt{13}+2\sqrt3-\sqrt7-\sqrt{19} \) and \( \varepsilon=2\sqrt{19}-5\sqrt3 \), for which \( 0<\varepsilon<\delta \). We also correct the range of validity of part (iii) of Theorem 4 of the same paper concerning unicyclic graphs, which fails at \( n=4 \). As an independent check, all statements are confirmed, within the stated ranges, by exact exhaustive computation over the \( 204{,}990 \) trees with \( 7\le n\le18 \) and the \( 1{,}039 \) unicyclic graphs with \( 4\le n\le10 \).