Fine knitting needles are needed for picking up and knitting off the finer machine-knitted fabrics. I've found a 0.7mm needle to be fine enough for fabrics up to 80 st/10cm. The fine needles come in short double points, or circulars; use circulars for wide panels, even if you're going to knit it flat. I found I could fit a sleeve on a 20cm double pointed needle (scrunched up) without having to worry about it dropping off!
Fine crochet hooks can be useful for picking up and knitting off fine fabrics. These can also be purchased on eBay - search for 'steel crochet hooks' and look for a set between about 0.50mm and 2mm.
Latch tools are used for reforming stitches when stitch-hacking. While very chunky latch tools are used in rug-making, those sold for machine knitting are a better size for stitch-hacking most knitted fabrics. They come in two sizes: chunky and standard.
If you are stitch-hacking a very fine fabric, a double-eyed transfer needle can be used to transfer the stitches from one side of the fabric to the other.
They can be bought on eBay - search for 'double-eyed transfer'. Like the latch tools, they come in different sizes. For fine fabrics, I would recommend the standard size, 4.5mm.
Throughout this re-knitting resource I have referred to many hand and machine knitting books. Some books are very useful for specific processes; others provide advice that is helpful for many aspects of re-knitting. Here are the top 5 books that I find myself referring to again and again:
Note: Many of the books I have referred to in this resource are out of print. I have provided links to Amazon listings for each book, to aid identification. I am trying to boycott Amazon, so would encourage you to buy books elsewhere - or to buy via Amazon Marketplace, rather than from Amazon itself. You can read more about the boycott and alternative sources here.