Gibbs505:
Here's a pictorial explanation of the cocking-piece/sear-engagement modification which is the likely cause of your problem:

The cocking-piece on the left shows the normal unmodified sear-engagement. You can see the shiny 'bruise' on the front face caused by the trigger-sear at rest when the rifle is cocked (1). Note that it is some distance above the lower edge of the face, which allows for the 2-stage pull common to many military rifles including the SMLE.

The cocking-piece on the right has been ground off below the sear-engagement to eliminate the 2-stage pull. It has effectively eliminated the second stage, so that the beginning of the first stage pull will now discharge the rifle. Note that the 'bruise' (2) is now immediately above the lower edge of the face, so that any movement of the sear will drop the cocking-piece and fire the rifle.

I'm betting that your rifle has been modified to remove the second-stage of the trigger-pull (as in 2). This is really only a problem if you want the rifle to remain original ie as a collector's piece. In that case, you should try to acquire a replacement cocking-piece which has not been modified.

Now, if you wish to eliminate the problem of catching the sear, you need only grind a small bevel on the lower lip of the half-cock notch and all will be well! This can be done with an ordinary grindstone without removing the cocking-piece from the bolt, but its easier to clean up afterwards if you strip the cocking-piece from the bolt first. In that case, you would need a firing-pin removal tool of some kind. I made my own.

Here is how to grind the bevel:

Here is a pic of a successfully-modified cocking-piece, showing the reduced sear engagement to remove the second-stage (1), and the small bevel which eliminates the sear catching at half-cock (2).

Hope this helps solve your problem, Gibbs505.