Put the mouse together and test it first before replacing the label and teflon sliders

Additionally if you want to make the job look professional you can replace the double sided tape and teflon sliders with a new ones, but cleaning old one off and using some superglue may do the trick

Using the opportunity of opening the case ie you can also give it a clean, the best cleaning solution might be IPA

The possible cause of this double click problem is probably used up connector inside of the microswitch or bend copper plate if someone used too much force on it, it might be possible to fix it by bending and cleaning it but there would be a risk that it will happen again shortly, it is better to replace it - it is also quicker