I will be up front and say that I am not familiar with NEC in respect to whether or not your GFCI is ok or not.

I want to say that it does not matter where the protection comes from as long as the protection is there, but I am not certain of it.

Personally, I would rather put separate GFCIs for the hallway and the bathroom. This is for convenience and to ensure that the user knows where the tripped GFCI is located. Imagine if your significant other, child, or guest ask you every time the GFCI trips (which hopefully is not often).

If you decide that you will GFCI protect your bathroom receptacle via the hallway, you must put GFCI protected on the bathroom receptacle. The GFCI you buy should come with a few. Since you have no ground, you must also put the no equipment ground label on both so that there is no confusion.

In the garage I have 3 receptacles daisy chained off one GFCI receptacle, but one can easily find the GFCI by following the conduit. For your bathroom it will not be so obvious.

The electrical shop in San Francisco where I bought my GFCIs sold spec grade Coopers for about $7.50 each.

Home Depot and Lowes sell Leviton and Cooper, respectively, for about $12 to $13 each. In the 3 packs, they charge about $30. I do not know if these are spec grade.

IMO, the difference in cost is not worth the inconvenience.

BTW, I like the Cooper more than the Leviton design if it matters to you. The Levitons have rectangular buttons that I find hard to press and there is a LED that is always illuminated unless in the tripped position. The Coopers have a square button with probably a bit more surface area and the LED is off until tripped.

If you have a grounding type receptacle and there's no ground it must be protected by a GFCI somewhere and marked "GFCI PROTECTED" and "NO EQUIPMENT GROUND." If the receptacle is the GFCI, you only need to say "NO EQUIPMENT GROUND" (the former is obvious). There's no requirement to mark GFCI-protected receptacles otherwise.

Doubtful he has conduit if he's got ROMEX in the circuit. The easiest way to find out if the GFCI is fed through is to push the test button and see if the other receptacles stop working.

If you decide to put in multiple GFCI's make sure you feed the second one from the LINE side of the first.

Thanks. I was concerned about the GFCI being able to protect the remote bathroom outlet that it fed by two wires...but it's already doing that local with two wires, so it's ok. I found the notes on the labeling.
Unfortunately I'm going to have to revisit.
One of the boxes is an old-work and is loose. The other is borderline too small for the GFCI. I'll replace with larger old-work boxes, but that is after I do some attic crawling and find the source of my no ground for this circuit.

Thanks

A GFCI device (breaker or receptacle) does not rely on the ground for correct operation, and is allowed under the NEC to be used to replace a receptacle on a 2-wire circuit.

The GFCI continuously measures the current on the hot and neutral of a connected load (wiring receptacles to the LOAD side of the GFCI receptacle extends this protection) and if the currents do not match within about 5 milliamps (0.005 amps) trips the GFCI to off.

The design works since it is rather rare for the current to manage to get from the hot complete;y back to the neutral without taking any other path back to the GFCI device and its sensor coil.

Using a GFCI provides almost the exact same level of protection as having a grounding conductor present.