The Critical Path is a mathematical analysis that identifying the sequence of activities that add up to the longest overall duration. In other words, this is the quickest the project can be done. However, this analysis does not take into consideration the criticality of the deliverables. It only identifies the longest (duration) sequence of activities, regardless of their importance.
Do not focus only on the critical path when you are evaluating which deliverables to watch closely. Also focus on your critical deliverables. A critical deliverable is a deliverable with a lot of risk, either by the impact, its likelihood or a combination of both. Some of these may or may not be on the critical path yet they may be more important for staying on schedule.
For example, a software project has the online help system as a part of the critical path. Nevertheless, the project can survive a late delivery of the online help system. This same project has the installation of a Server as deliverable with a large float associated with it. It is not on the critical path, yet if the server is not ready in time, the effect to the project is very high. It is sensible to make extra efforts to make sure that there are no delays to the Server.
This is risk management and managing risk is not an isolated activity. It is a part of many project activities, including schedule management.
Remember there are other factors to consider when identifying items for which a timely delivery is critical. Remember the critical deliverables.