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Can I have something like this, or does the cable run from the panel to the subpanel need to be dedicated?
Main Panel+Breaker -> 40A dryer box with Outlet -> Subpanel+Breakers -> More circuits?
Note that I'm not plugging the subpanel into the outlet! Everything would be hard-wired, I just want to know if an outlet can exist on the same circuit that also feeds a subpanel.
A feeder is breaker and wiring that supplies a subpanel.
A branch circuit is breaker and wiring that supplies various outlets, including receptacle outlets.
"Outlets" does include hardwired loads, or as they like to call them, "utilization equipment".
Wiring can be one or the other, not both. You cannot have outlets on a feeder.
So no.
The only exception I can think is if you have a reason allowed under 400.7 or 400.8 to have the subpanel be fed from an inlet or flexible cord plugged into an outlet. (yes, I know you said you don't mean that). For an example of pretty much this, look at a PDU as found on a computer rack. A PDU is basically a rack mount subpanel. However, your plan would require multiple receptacles - one for the subpanel and one for your load. The largest circuit which allows multiple receptacles is 30A. So that would limit usefulness of your subpanel.
The normal solution is to run a, say, 50A feeder to the subpanel, then have a 50A branch circuit fed off the subpanel. That is legit if the load calculation for the subpanel says that will work. Which it might if the other loads are just incidentals like lights and receptacles (e.g. circuits that allow 180 VA allotment per receptacle). You can't plan to overload a subpanel.
First of all, a dryer receptacle is rated 30 amps, not 40, so if it's on a 40 amp breaker, you're already in violation.
If you truly have a 40 amp circuit (#8 copper wire minimum) the you're ok for the wires being on a 40 amp breaker but not the receptacle. If you have #10 copper on a 40 amp breaker then you are overfused.
Either way, there is nothing wrong with removing that receptacle and replacing it with a sub panel. If you're planning on still using the dryer and you have a 30 amp circuit and not 40 as you stated, then you probably don't have much headroom to additional loads imposed on that circuit as a dryer almost certainly will consume close to 80% of 30 amps. If you truly have a 40 amp circuit (#8 wire) then you could conceivably swap the receptacle for a small panel, then put the dryer receptacle on the new panel (this time on a 30 amp breaker and #10 wire) and add additional circuits to the panel. You wouldn't have much headroom though to add a whole lot. Maybe a few convenience receptacles and/or lighting circuits.
If you have #10 wire and it's in conduit, you could pull larger wire, say #8 or #6 to the new panel. I think you can get 3 #6 in 3/4 EMT with no ground wire (use pipe as ground).
If wired the way you suggest the breaker would have to protect the outlet and the entire subpanel. The breaker and the outlet would have to match. There is no useful combination of breaker/outlet/subpanel that could work this way.
You can't run say a 60A subpanel off a 60A breaker and put a 20A outlet on the feeder cable.
If you use a 15A breaker and a 15A outlet, you don't need a subpanel, you could just run more 15A outlets along the same circuit. Same with 20A.
If you use, say, a 50A breaker the outlet must be 50A. Then, if it's allowed you could continue to a 50A subpanel. (I don't think that's allowed anyway). You could only attach a 50A device to the outlet and you could only use it if nothing was running off the subpanel. The whole arrangement would be pointless.
If you use an even bigger breaker the outlet would have to be correspondingly bigger and then there is literally no domestic device you could plug into it.
You could run the subpanel conduit through a box (it would have to be a larger than usual box) and install an ordinary outlet there but not connected to the feeder cables. Feed the outlet through wires coming back from the subpanel through the same conduit. Would that suit your needs? The need for the box to be larger, and to accommodate the larger conduit meant for the subpanel, might spoil your plans, depending on what they are.
