From my experience, talking about the B737 and upwards and similar types, the teaching has always been that if a problem occurs then select the auto-pilot 'in' so that you can sit back and properly assess the problem(s) and make the correct decisions.
Very few people can do two things at once so the idea that if things go wrong you should hand fly is not, in my opinion, a good one, unless the nature of the failure prevents the use of autopilots.
Possible exceptions would be an engine failure at or just after V1, then I would expect the PF to hand fly until the clean up was complete but even then selecting an auto-pilot 'in' at a safe height, (say 500'), would not necessarily be a bad move.
In todays two crew environment with highly sophisticated automation on the flight deck I personally believe it would be foolhardy not to use all the systems available to your advantage.