You do not need a hash. In fact, your original code is very close to doing what you want. Just a few tweaks:

#!/usr/bin/perl
use strict;
use warnings;

open(FILE2,">file1.txt")|| warn "Could not open\n";

open(FILE3,"file2.txt")|| warn "Could not open\n";

my $Previous = "";
my @data = <FILE3>;

    my $index=0;
    foreach my $_data (@data) 
    {
        $index++;
        chomp ($_data);
                my @Current = split(/\s+/, $_data);

        if ($index == 1)
        {
            # do nothing.
        }
        else
        {
            my @Previous = split(/\s+/, $Previous);
            if ($Current[0] ne $Previous[0])
            {
                print FILE2 $Previous, "\n";
            }

        }

        $Previous = $_data;
                 
        }

    if ($Previous) {
        print FILE2 $Previous, "\n";
    }

close(FILE2);
close(FILE3);
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I made the following changes:

• Added strict and warnings. Corrected @foo[0] to $foo[0] (see the perldiag entry for the warning it gave before), and declared variables.
• Moved the $index check so $Previous isn't used unless it's been set.
• Changed to split on whitespaces, not tabs (since the data you provided didn't have tabs).
• Add newlines to what's written out.
• Add block after the loop to print the final line that had been saved.