First, you say your service is 100A, but those "main supply" wires look awfully small for 100A. They look a lot like #6 copper, which is only good for 60A tops. I would make sure the wires are appropriate for the main breaker. If they are inadequate, either upsize the wires or downsize the main breaker.
This ... I hesitate to use the word "service panel" ... is still supported. Bryant was bought by Cutler Hammer, in turn bought by Eaton, and became Eaton's "BR" service panel line. "BR", Bryant, get it?
Yes, you can get a "quadplex" double-stuff which fits in two spaces and supplies two 240V circuits. This quadplex will have proper and internal handle-ties, NOT A FLIPPIN' NAIL. For instance this quadplex includes a 2-pole 40A and a 2-pole 20A. It would replace the breakers with the nail.
That breaker isn't just two twin breakers bolted together. It is purpose-built to be a quad. Internally there is a mechanism to assure both sides of the circuit trip together if either overloads (common trip). That works with or without the goofy looking handle-ties. The handle-ties are just UL listed versions of a nail; they are there to provide common maintenance shut-off. With a nail (or listed handle-tie), common trip is not guaranteed.
You cannot add any more breaker spaces to this panel. The cover has additional knockouts, but there'd be nothing under the knockouts. This is an 8-space/16 circuit panel, you are using 14 and have only 2 more possible.
After this you can add 2 more 1-pole circuits or 1 more 2-pole circuit and your panel is wedged solid. You might want to think about a bigger panel at some point. Get a panel so large that you never, ever need to use double-stuff breakers. (because increasingly, they are requiring GFCI and AFCI breakers, and those don't come in double-stuff.) You need 14 now, 24 is borderline too small, I would go at least 30, even 42.
You don't need to deprecate that panel, you could add a larger panel as a subpanel fed by a large breaker.
You could stay in "BR" panels if you really wanted to, it is still common and supported, though a bit on the cheap side. Siemens is commercial grade (so are CH and QO, but their breakers are rather expensive).