You may have noticed that the only link to this page was at the end of the last of all the other mail pages. I reckon that if you have got this far, then you have shown the dedication which earns you the right to learn a bit about riveted mail. The topic is as big as the rest of mail making, and so I shall only touch on it, until one day you find me meditating on a mountain top, and you show me your armourers' secret tattoo, and all that.
To cut a long story short, to make riveted mail, you will need to manufacture the tools for the job, and these will have to be made out of very hard metal indeed, much harder than the metal of the links. You will need to case-harden them in a forge. The first of the two main tools is a substantial hinged thing into which you place a ring with overlapping ends. The ring sits between two dies which hold it in place, and impart their negative shape to the ends of the ring when you bash the top half of the tool with a big hammer. You then open up the tool, and see that the ends of your link are now squashed out flat, in a characteristic "watershed" shape, which is elliptical, and thicker in the middle than at the edges. The second tool is for punching holes in the flattened ends of the links, once the link has been linked into position on the shirt. It looks like a big pair of tongs, and has a pair of dies in its business end. One has a spike on it, and the other a receiver for this spike. Remove the dies (or have a second pair of tongs) and replace with dies shaped for closing the rivets. Insert a rivet, and crush it into place with the tongs.
The above does not mention how to make the links in the first place, nor how to make the rivets. The rivets would be made by an assistant in the armourer's workshop, from wire cut and bashed into tiny wedges. Decades of fun. The wire would be made by drawing metal through steadily smaller and smaller holes in a block of metal, using a hand-winch. The wire would then be wound by hand round a metal rod, then the coils would be cut with a chisel into rings, which would then be hammered using a special rod with a shaped end, through a metal funnel, and then they would fall out of the funnel with their ends overlapping by just the right amount.
One of the reasons that mail was riveted, was that the iron used for the mail was quite soft. Modern tough springy metal is more suitable for butted mail. However, once the shirt was finished, the whole thing could be packed in carbon (charcoal) and case hardened in a forge (but a mistake in this process could turn it very brittle). A ring which is linked to itself with a rivet is stronger than one without a rivet, and so protects the wearer more. However, some people argue that because it was stronger, it could be made lighter by using thinner wire for the rings, but thinner wire would be weaker. I am not sure how real armourers used this trade-off. Did they make mail all of the same protective quality, but varying in weight, or did they make mail varying in protective quality, but of the same weight? I have read that the former is the case amongst some modern makers, but my suspicion is that the latter is more likely for the ancient and medieval world. All types of armour, scale, lamellar, brigandine, plate, mail, tend to weigh about the same. It seems to me that there was an amount that a man could wear without weighing himself down too much, and armour was made to that weight, for maximum protection.
So, now you know a bit about how you can make riveted mail.
Or you could just go down the pub.