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PLEASE start becoming exact and consistent when posting your problems here, also across posts, making samples match what you say in the text.
People (not only) in here tend to refer to the...

you might/will run out of the opened file descriptors.
Here's a slight variation on the theme:

awk '
FNR==NR {f21[$1]=$3
f22[$2]=$3
next
...

With a VERY sloppy interpretation of "along these lines" you might come close to the desired result, once you corrected the syntax / redirection error in the before-last line, and accepted the higher...

to start with:

awk '
FNR==NR {
f21[$1]=$3
f22[$2]=$3
next
}
$1 in f21 { print $0, f21[$1];next }
$1 in f22 { print $0, f21[$1] }
' file2 file1

Still not unambiguous. Try
awk '
NR == FNR {T[$2] = $0
next
}
($2 in T) {print T[$2], $NF
}
($3 in T) {print T[$3], $NF
...