How can I make this necklace longer?
If you want it to hang vertically you will need to do a fair amount of reworking of the setting, this is complicated by any soldering runs the risk of damaging the stone. 

If you just want to extend the chain it is probably just as easy to buy a new chain of the correct length and do a direct swap. You can get a huge variety of chains from jewelers supplier or indeed ebay for not too much money, indeed just a waxed cotton or leather thong might be just as good with this piece. 

If you want it mounted vertically you will need to cut off the existing rings and file them flat to the mount and then add a new ring at one of the narrow ends. 

Here you have two practical options, first to get a blank silver mount of the right size and shape and glue the existing piece on with epoxy or you could wrap the whole thing with silver wire to create a sort of 'cage'.