Your method requires 2 full passes of the file, and then creates the problem of trying to relate the results of the two.

Far simpler would be to do a single pass, remembering each section and discarding it when you see the start of a new one. And then print the remembered section when you see the search term. You have a single pass, that on average will complete half way through the file, and no silly games trying to relate two separate searches.

Ie:

#! perl -sw
use strict;


my $term = shift;

my @section;
while( <DATA> ) {
    @section = () if /Header/;
    push @section, $_;
    if( /$term/ ) {
         print for @section;
         print while defined( $_ = <DATA> ) and not /^Header/;
         last;
    }
}

__DATA__
Header 1
just some junk
just some junk
just some junk
just some junk
just some junk
Header 2
just some junk
just some junk
just some junk
just some junk
just some junk
Header 3
just some junk
just some junk
just some junk
this is a term
just some junk
just some junk
Header 4
just some junk
just some junk
just some junk
just some junk
just some junk
Header 5
just some junk
just some junk
a different term
just some junk
just some junk
just some junk
[download]

A couple of sample runs:

c:\test>junk34 term
Header 3
just some junk
just some junk
just some junk
this is a term
just some junk
just some junk

c:\test>junk34 different
Header 5
just some junk
just some junk
a different term
just some junk
just some junk
just some junk
[download]

Note that this stops looking the first time it matches the term. To display all sections containing the term, just delete the last statement.