http://www.cyclespeugeot.com/images/scan0009.jpg shows that the bike has a freewheel on threaded hub.
I think 6 speed indexed down tube shifters is a very useful set up - simple, usable and reliable.
However, 13-21 is what people raced on. Everyday utility riding was done on 14-28 - or 14-24 if you wanted closer spaced gears. I'm going to tell you how to change the freewheel.
Obtain the correct type of freewheel remover - if it is a shimano freewheel, you need the shimano kind of tool - I recommend Park Tools as a brand, you will also need a one inch spanner, although a large adjustable spanner will work.
Using the freewheel tool and the spanner, unscrew the 13-21 freewheel. If it won't budge, squirt some penetrating oil at the threads and leave it overnight then try again. It will come off eventually.
Replace it with a 14-24 freewheel. It should work with the derailleur you have. This will give you a bottom gear 14% lower than you currently have.
You will probably need to add chain links, and you may need to replace the chain if it is worn. This will involve obtaining a chaintool and a new chain. Six speed chain is inexpensive and highly durable.
If you need lower gears still, you can opt for a 14-28 freewheel for a 25% reduction, but that probably won't be necessary.
(A 39/52 chainring or a triple chainset will make a slight difference but the triple may require a different front derailleur to allow you to access the small ring, may also need a different bottom bracket which will require a special tool for the removal of cranks and one for the removal of the BB. So, don't worry about that until you've sorted the freewheel - which will probably give you all the gearing you want.)
(Cyclofabrica, the main problem with changing to 7 or 8 speed is axle width incompatibility, besides the need to replace shifters - DT friction is a chore.)