liam.zbasic wrote:... Realistically however, my project will only produce values from say 2.0 to 2.5V...
Two methods to increase sensor signal resolution are offset amplification, and signal processing. The first requires additional hardware - usually, at minimum, an opamp and surrounding passive components - while the latter requires time. The first works best if the signal is noise-free, while the latter requires some noise to dither the LSb.
The simplest method, assuming a signal with at least 0.5bit of white noise, is averaging. Using averaging to improve resolution is an attempt to extract signal
below the noise level or to interpolate data inside one bit. Averaging takes time so the signal bandwidth (how quickly the device can respond to a change) is reduced. To statistically double the resolution, i.e. to add one bit of resolution, four samples are required (the square of the resolution improvement), so the process slows by a factor of four and the bandwidth of the signal is quartered; a two-bit improvement (four times resolution) requires 16 samples and is correspondingly slowed. Resolution improvements become rapidly more costly in time - and the result becomes increasingly untrustworthy as the qualities of the noise become more dominant. More sophisticated signal filtering methods are available, too, but they are not trivial.
An opamp can be used to both subtract an offset and add gain to redistribute the signal dynamic range over the available ADC range. In your case, you can subtract 2VDC from the analog signal and multiply the maximum-signal difference (0.5VDC) by 5.0/0.5= 10.0 (the gain) to feed a 5v-reference ADC. The best result you can achieve with a 10-bit ADC with a 5v reference and a 2.0-2.5v signal range, then, is a 10x resolution improvement, a little more than three bits. That would be the equivalent of a 100-sample average if the noise were white. Signal noise is also increased by the gain, though, so a noisy signal might not be improved. Simple DC opamp circuits can suffer from instabilities, like temperature sensitivity, that limit the accuracy of the solution so a proper design could be involved.
You might find that some combination of hardware offset/gain and averaging will produce the best result, but if you need more resolution than a few additional bits, an external ADC is probably required.