Depending on what you mean by "continue", it might be best to locally override CORE::GLOBAL::die, like so:

{
    my $realdie = \&CORE::GLOBAL::die;
    local *CORE::GLOBAL::die = sub {
        $realdie->(@_) if $something;
        print LOG @_
    };
    ## do_stuff
}
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See the section on %SIG in perlvar for some discussion.