If you've got an Intel chipset you could take a look into combining the HDD and SSD into one volume if you have Intel Smart Response Technology.
It's something that's best done when you're setting up a system though, as what you do is set everything up normally on your HDD, then enable the feature in order to automatically use the SSD as a cache (wiping the SSD); this means that most of the time all your data is cached onto and coming from the SSD, meaning the HDD can just spin idly (or spin down in some cases) which reduces noise from it considerably. If you've got the feature I highly recommend enabling it, but it depends on the motherboard you have. If you've already set up your system with files separated between the SSD and HDD then it's more complex to decide how to separate them out.
Of course that's contingent on you having the feature, a lot of motherboards do though so it's worth a look. Otherwise like others have said you can either get anti-vibration screws, or even just rubber washers in the right size to add to the screws you have. A lot of hard-drive noise isn't from vibration alone though, some drives are just noisy, it really depends on whether it's noisy during light use (probably vibrations), or only when loading large files (in which case it's a noisy drive). There are other things you can do too, for example you can buy noise-dampening matting to put inside the case, which cuts out some of the harsher sounds that can be distracting, but it's expensive for what it does.
Alternatively consider getting another HDD; if you're currently using a 3.5" drive then you might consider a 2.5" drive if your case has a suitable mount, these tend to be a lot quieter, but are also slower for random access (you wouldn't want to boot your system from it unless you're using Smart Response)
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