Yes, in theory you can do what you want, but only if you have some wholly unrealistic equipment available to you.
The several other comments made so far about limited extra accuracy are correct, alas.
Measure a voltage with a 12 bit ADC and get say 111111000010
You know that the real value lies somewhere in a 1 bit range +/- 0.5 bits either side of this value.
IF your ADC was accurate to 24 bits but was providing only 12 bits then it is reporting that the vaklue lies within +/- half a bit of
111111000010 000000000000. If this was the case you could take a 12 bit ADC with a +/- 1/2 bit range, centre it on 111111000010000000000000 and read the result.
This would give you the difference bwteen the actual signal and the aDC value, as desired. QED.
However the 12 bit ADC is itself only accurate to about half a bit. The sum total of its various errors cause it to declare a certain result when the real result is up to about half a but different plus or minus.
While you would like
111111000010 to mean 111111000010 000000000000
it may actually mean 111111000010 000101101010 or whatever.
SO if you then take a 2nd ADC and measure the lower 12 bits and ASSUME that they are relative to an exact 12 bit boundary, they are actually relative to the above erroneous value. As this value is essentially random error, you would be adding you new 12 lower bits figure to 12 bits of essentially random noise.
Precise + random = new random.
Use two conveters that can measure a range and give a result in 1 of 10 steps.
If scaled to 100 volt FS they give ge 0 10 20 30 40 50 60 70 80 90
If scaled to 10 volt full scal they give 1 2 3 4 5 6 7 8 9
You decide to use these two converters to meaure a 100 volt range with 1 volt accuracy.
Converter 1 returns 70V.
You then measure the voltage relative to 70V and get -3V.
So you conslude that the real value ie +70V - 3V = 67V.
HOWEVER the 70V result could in fact be any of 65 66 67 68 69 70 71 72 73 74
Only if the 1st converter is ACCURATE to 1V in 100, even though it displays 10V steps in 100V, can you achieve what you want.
So you real result is 67V +/- 5 volt = anything from 62V to 72V.
So you are no better off than before. Your centre has moved but it may be located randomly.
You will be able to get modest improvement this way as a converter is usually probably slightly more accurate than the bits it returns (you hope) so your 2nd converter makes some use of this.
A system that does in fact work has been mentioned with one important omission.
If you sample a signal N times and you add +/_ half a bit of gaussian noise you will spread the signal "all over the possible range" and the average value will now be log(N) more accurate than before. This scheme has fishhooks and qualifications and you cannot just get an arbitrary extra number of bits, but it does offer some improvement.
In the first case above I mentioned a12 bit ADC with 24 bit accuracy. You can achieve something of th sort by using a 12 bit ADC and reading its assumed value with a 24 bit eg delta sigma converter. IF the signal was stable enough that it remained in the same one bit range you can use a 2nd ADC to read the 2nd 12 bits wrt this stable signal.
Alternative - just read 24 bit signal initially with sigma delta, lock in that point and then mesure successively relative to it with the 2nd ADC.As long as the signal stays within range of the 2nd ADC you'll get a much faster result.