The best and fastest way to get a high pairing when Klonoa is already at max walking speed (and that this speed would not give a high pairing) is often to stop moving for one frame (which sligthly decreases Klonoa's speed) and then start moving again until a high pairing is available (after 4 or 5 frames usually). The highest pairing you can get is 53248-55296 (so a speed of 54272 on average), which is ~1,6% faster than walking, saving on average 2 frames every 3 jumps.

While randomly jumping would usually result in a slower speed than just walking, jumping at specific speeds in order to get a high 'aerial speed pairing' is actually faster than walking. It means that once Klonoa reaches his aerial top speed, his speed will go up and down by 2048 until he is back on the ground. If his speed is higher than 53248 (0xD000), then it will decrease by 2048 instead. In the air, Klonoa's speed goes instantly to 18432 in 1 frame (as long as his previous speed was lower than that), and in other cases increases by 2048 each frame. If his speed is higher than that, then it will decrease by 512 every frame until it reaches ~53250. On standard ground, Klonoa's speed goes instantly to 41369 in 1 frame (as long as his previous speed was lower than that), and in other cases increases by 409 each frame until it reaches ~53250.

Klonoa door to phantomile emulator