Certainly sounds like a worn 2nd gear syncro. They're usually the first to go because it is the most used change and one with a large speed differential.
Double clutching will help overcome the problem (more so than just a blip on the throttle).
The technique is: press the clutch in, shift to neutral, let the clutch out, bring the engine revs up slightly to match the next gear speed, press the clutch in with the revs up, select the gear, let the clutch out.
It takes quite a lot of practice to do it quickly and get it right. But it's how everybody changed gears in the pre-syncro days. Unless of course you had a Pre-selector gearbox like in the Armstrong Sidleys - but that's another story.

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Curly
'64 Cooper 'S' (Group Nb spec)
'67 Mini Marcos GT www.mm7056.wordpress.com