I don’t see any reason to think that there is a nice expression for the new RHS. Suppose that $a$, $b$, and $c$ are the smallest of the six numbers; then the RHS is $a+b+c-\min\{a,b\}-\min\{a,c\}-\min\{b,c\}+\min\{a,b,c\}=\max\{a,b,c\},$ as before. If, on the other hand, $a$, $b$, and $c$ are the largest of the six numbers, then the RHS is $a+b+c-d-e-f+\min\{d,e,f\},$ so it’s one of the numbers $a+b+c-d-e$, $a+b+c-d-f$, and $a+b+c-e-f$, none of which need have much of anything to do with $\max\{a,b,c\}$. Other relationships between $a$, $b$, and $c$ on the one hand and $d$, $e$, and $f$ on the other lead to even more variety.