I use a Canon 70-200 mm f/2.8L IS II USM that I primarily use for nature photography, but because of the focal length it is also ideal for this purpose as well. The set up that I use is to have the camera about eight feet away from the subject which means that the FL (focal length) will generally be in the range of 80 to 120 mm. For a DSLR this seems ideal because it closely emulates human eye perspective from a typical viewing distance.
While having a fast lens (meaning a large aperture) is desirable for many other types of photography, you definitely do not need a fast lens for this type of product photography. In fact just the opposite is the case because you will need to stop the lens aperture down quite a bit ... approximately f/11 to f/16 to gain sufficient depth of field so that everything is in focus. That is actually a good thing because large apertures mean a heftier price tag.
Many other woodturners use wide angle to normal zoom lenses, large aperture, and shooting up close. The problem that I see with that approach is considerable perspective distortion especially at the shorter focal lengths (the "fat nose" effect).
There are several affordable standard to medium telephoto zoom lenses that will work, but you are right that the kit lens that came with the camera while not terrible also isn't great. Just don't let price be the primary deciding factor. Zoom lenses are always a set of design compromises and two things exacerbate the compromises -- low price and wide zoom range. The price part is probably self-evident because a $200 lens isn't in the same league as a $1000 lens. Zoom lens designs have to make compromises in sharpness of focus, astigmatism, and chromatic aberration. Usually, a wide range zoom will be sharp at the shorter focal lengths, but not sharp at the long end of its zoom range. I used to own the
EF 75-300mm f/4-5.6 III lens which is Canon's lowest cost lens in this category. It is OK except at the long focal lengths focus isn't sharp and it has more than its share of chromatic aberration at the long end. It shouldn't be confused with Canon's four 70 - 300 mm lenses that have better performance (and cost more).
Here is a link to a Canon EF-S zoom lens with a focal length range of 55 - 250 mm --
EF-S 55-250mm f/4-5.6 IS II. The list price is about $250. While it is towards the lower price end of Canon lenses, I have heard good comments about its performance. Generally the performance drops off at the long FL end of these lower cost lenses which isn't really surprising. Note that EF-S lenses will only fit on Canon's APS-C format DSLR bodies just in case you are contemplating going to full frame sometime in the future.
Another especially good zoom lens with a very good price considering its performance is the Canon
EF 70-200mm f/4L USM lens That sells for about $600.
Don't forget that you will need a tripod ... a good sturdy one and a way to remotely trigger the shutter if the camera doesn't' have a built-in timer.