For minimum phase noise/clock jitter, would it help to solder the can to GND?
It cannot do much harm - use a fine wire, and lead based solder, so you can use least heat.
Nothing metrology grade, weekend tinkering but with a need for low phase noise.
If you chase phase noise, other things may dominate.
MCU oscillators are forced to share the GND and power pins, so they have significant MCU activity crosstalk.
Low ESR certainly helps. If your crystal was from a HDD, it was likely cheap, rather than high spec, but it should have 'aged' by now
You may be better
using an external oscillator, either one spec'd for low phase noise, or you could try a crystal with this part :
https://www.nisshinbo-microdevices.co.jp/en/products/quartz-crystal-oscillator-ic/spec/?product=nju6385That is a Xtal Oscillator that is specified to have low phase noise.
Nisshinbo have other (cheaper) siblings, but that one is spec'd to have low phase noise, and they give the Xtal spec's ( ESR = 17.7 ohms).
Abracon do spec a special low ESR crystal, in the ABLS-LR series (same large HC-49/US case as your xtal), but they are somewhat rare and no stock shows for 20Mhz
Mouser have 16,24,25MHz models, with the 24MHz one done in lower ppm.
It would be interestieng to compare a 'best discrete' solution like (say) NJU6385 + ABLS-LR-24.000MHZ-B-1-G-T
with a 'good' oscillator (this one 24.576MHz )
https://www.lcsc.com/product-detail/Crystal-Oscillators_Seiko-Epson-X1G0041710029_C255994.html ( 40c / 100+)
Addit: If you really want low phase noise discrete designs, I found this link that shows how important crystal variations are
https://www.diyaudio.com/community/threads/the-well-tempered-master-clock-building-a-low-phase-noise-jitter-crystal-oscillator.261651/page-150#post-6385649If you are making these in higher volumes, I saw recently lcsc have a new category.
https://www.lcsc.com/products/Temperature-Sensing-Resonators_13490.htmlThese parts are designed for high performance oscillators, so they have low initial ppm and low aging, and high min Q ( 75,000 ) & come in 2520 packages.