I would, i bled the system with my wife's help rather than the exchange method.

 

2nd gear syncro is triple cone. All other syncros are double cone.

This means you have an extra clutch face on the syncro for 2nd gear which is part of the reason that it is notchy when cold.

You have the extra cone because the speed differential between the gears is greatest between 1st and 2nd gear but the actual gear speeds are lower so it takes more pressure on the cone faces to enable the gears to match speeds. Hence 2nd gear syncro does the most work in the box.

Changing from 1st to 3rd when the gearbox is cold is probably the worst thing you could do for the gearbox unless you double declutch or take it very slowly and if you're making the effort to do that you may as well go through 2nd gear and not rev the engine as hard when that's cold.

 

The extra cones should make it less notchy, as it should synchronise faster - provided the right oil is used, if the oil is old or too thick, there's your issue, 3 times as much film thickness to push through...and if it has EP additives or a poor friction add pack, even worse. I'm not sure what it'd be like in the S2000 'box, but in some other cars known for poor shifting in 1-2nd when cold, that also use a pretty thin MTF like the Honda OE fluid, Syntrans B makes a good improvement in change quality...