That assumes that the starts are on consecutive lines which may be a perfectly valid assumption. It pays to know your data.
Another approach is to use the original regex, which may have multiple starts and then trim it using s/^.*start/start/is.
The loop then looks something like
while ( $string =~ /(start.+?end)/gis ) {
my $data = $1;
$data =~ s/^.*start/start/is;
print $data, "\n\n";
}
If the intent is to strip off multiple starts only on consecutive lines then the regex would be
s/^(?:start\s*)+start/start/is which used on the input
start
start
start
go
one
end
start
start
data
start
go
two
end
would produce
start
go
one
end
start
data
start
go
two
end
But as I said you really need to know your data and other factors such as if you need to validate the input.