Intermittent problem starting the car

Hello. The problem is that the car, occasionally (can take up to 2 weeks per occasion - totally random) will not start without a push start.

I have tried the following:
Switching on and off the alarm system
Using the key to lock and unlock the car n case it affected the alarm system
Constantly tried to turn the engine over - but after a short period (20 seconds or so) the starter motor stops. I retry and it stops after about 3 seconds.
Tried starting with short bursts of turning the starter motor over.
Replaced the battery last October - fine of 3 months then started again. Recharged the battery yesterday, put the battery back in yesterday and it wouldn't fire up this morning (but the starter motor turned over ok).
Checked the oil.
Happens rain or shine, hot or cold.
Starter turns over, but no sound of the engine wanted to fire at all.

Only ever happens after the car has rested over night or during the day for 8 hours (whilst its parked at work)
Always starts after a push start - usually within a few feet.

Today I learned a new way of starting it: I engaged the gear, turned the key and the jumped forward as expected. I then took it out of gear, turned the key but it just turned over. So, I put it in gear again, turned the key and the car jumper forward. I took it out of gear and turned the key and hey presto it started!

Any ideas would be appreciated.