Sounds similar to a problem I had. When cold the car would engage 2nd gear (normal) on a reasonable quick throttle, then jump back to 1st gear, tear away and then give a neck jerking change to 2nd. When hot it always started in 2nd gear no problems. I suggest you look at the rev counter to check which gear the tranny starts in when cold. 1st gear should spin happily to about 40mph at 6000revs and provide very rapid acceleration. 2nd gear starts on the 300's I've tried are sluggish which is why I bought a 320E- WOW.
To recap I suspected a sticky component in the valve body although can'be sure. I changed the oil and filter in the tranny and I've had no problems since.
1993 320E 140,000km