I don't suppose fixing whatever's creating those logfiles is an option?

Here's what I might do: remember the previously-read line, and match your search term against both the current line and the concatenation of the previous and current line. Off the top of my head (completely untested):

my $previous = "";
while(defined(my $line = <>)) {
    chomp $line;
    ($previous . $line) =~ m/$pattern/ and say $line;
    $previous = $line;
}
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(This isn't perfect, obviously; it'll also print lines where the previous line just so happened to match, even though there's no match across the line boundary. But it should get you started.)