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^^^ Exactly, by double clutching, you reduce the loading on the syncronizers to the point where they do not have to do all the work in bridging the speed difference between the layshaft and the transmission.
Since the noise goes away when you double clutch - you have just narrowed down the possible suspects to excessive syncronizer wear (very probably) and possibly bad gear oil.
Flashmn brings up a good point about GL fluid ratings. You can buy GL-4 and GL-5 gear oils. Though many are cross-compatible (according to the fluid manufacturer), there is quite a bit of controversy about how corrosive the GL-5 fluids are to certain metals (so called yellow metals) in the transaxle. You can guess where this is leading to - as almost all syncronizers are made from brass - there is some concern that running the wrong spec'd fluid will cause premature wear.
The same can be said with trying to shift aggressively or "rev" match the transaxle - done incorrectly over a period of time, will eat the syncronizerd pretty quickly.
Quick, easy fix - change the fluid with OEM spec'd fluid - should be marked in the owner's manual and Factory Service Manual. If it says use GL-4 oil, use only that. Don't be tempted to add any additives right from the start - you need to change the fluid first - then see if there is any change. If not, then try an additive. Stacking multiple additives will not fix the problem faster, if anything - they could potentially mask problems and you'll have a more expensive fix on your hands. Fluid exchanges will help, but will not fix any mechanical issues (wear and tear) inside the transaxle.
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2002 Corolla S, 1.8L 1ZZ-FE VVT-i
2003 Matrix XRS, 1.8L 2ZZ-GE, VVTL-i (RIP)
2009 Matrix XRS, 2.4L 2AZ-FE VVT-i
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