Can a Variac be modified to output higher voltages?
If I want my Variac to output between 0 V and 415 V when fed with a 240 V input, am I correct in assuming that all I would need to do is move the mains input wire to roughly 60% along the coil? (240 / 415 = 57.8%).


No. You are completely incorrect. 

This is just another version of 'can I use my step-down transformer backwards to step up'?

Any transformer core of a given cross-section running at a certain frequency has a maximum 'volts per turn' it can produce before saturation. You can safely assume that your transformer's 240v tapping point is already at as low a turn count as it will safely go. 

If you do tap at 60% of the 240v turn and connect the transformer to 240v, the fuse or breaker will go. If you replace either with something that will handle 10x the current, then your transformer will smoke briefly before failing. Transformers rely on the primary inductance to keep the magnetising current negligible. Once the core saturates, the inductance collapses, and the magnetising current becomes huge, invariably enough to overheat the windings.

The way using the smallest amount of material to get 415v from 240v is to get a 240v to 415v autotransformer. An alternative if you can't find one of those is to use a 240v to 175v step down transformer, and put the secondary in series with the primary supply. Depending on phasing, you will either get 65v or 415v. Once you've connected it to get the latter, you've built a home-made 240-415v autotransformer.