Sounds as if the safety spring (left side of the rifle, it`s the strap-sort-of-thing with the screw at the bottom) either is weak or not bearing properly. This is somewhat of a common problem.
The actual problem could be either the spring has lost its springiness, or it has gotten bent somehow, or it has come loose. There should be a rather solid feel to the safety and it shoundnt go flipping itself off.
First, check if the screw is loose. If it is loose, tighten it up and try the rifle.
If the problem persists, undo and remove the screw, then lift off the spring itself, being very careful not to remove or disturb the safety mechanism. This thing uses an 8-start thread and needs to be timed in order to work; by NOT removing it, you won`t need to re-time it. Check for dirt and garbage under the spring. If there is some there, clean it out.
You can ADD PRESSURE to the spring by bending it slightly farther than it already is bent. Then you put it back on and try the rifle. Chances are the problem will be gone.
There is always the possibility that the spring has lost its temper (I do this all the time!). If this has happened, the problem will recur. In this case the remedy is to replace the safety spring with a new spring. They are only worth about 3 bucks, anyway.
BTW, the Lee-Enfield action is fast enough that you are taking a silly risk by carrying the thing with a round up the spout, right next to your ear. Try repairing the safety, then carrying the rifle with the chamber empty. Be a lot safer. I lost 40% of my hearing to a single gunshot (big one, mind you, but I was yards from the muzzle; carrying a rifle like you are would guarantee a burst eardrum if it fired), really don`t want to hear of people getting in the same fix. Be safe.
Hope this helps.
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