Would it offer any additional level protection if I put it in a
static-shielding antistatic bag whenever I carry it around (e.g.
before putting it in a bag or cardboard box)?
Yes, you do get additional protection.
Exposed potentials (such as those on connectors) can be a conduit for ESD. If a potential is developed (especially the thousands of V/m that can be developed from clothing) across the part then connectors such as USB, or Ethernet can be affected. An ESD bag will prevent these electric fields from affecting the inside of the bag. (make sure you don't use the polypropolene (or pink bags) as they are intended for shipping, and the conductive layer wears off, use the mylar metalized bags).
That being said the connectors are partially connected by the shields on the outside of the conductors, an electric field or ESD event is most likely going to conduct through the shield (which is traditionally connected to ground on the PCB).
It really depends on how much the product is worth protecting. The probability of you degrading an enclosed PCB is very low, but it's not zero. If you value it then protect it in an ESD bag.