Can I speed up the experiment even further by choosing a faster heating rate?

Well, the heating rate should not be too large because there is always a temperature lag between the target temperature and the actual sample temperature.

Basically, heat transfer between the holder and the sample is limited. This depends on the sample cell (and, if applicable, adaptor) dimensions (e.g., wall thickness) and material, the sample itself, and the temperature gradient between the two.

Due to this limitation, usually two effects can be observed:

The sample temperature will not always catch up immediately with the temperature of the holder, i.e., the target temperature.
The sample temperature remains somewhat below the target temperature even if the system has equlibrated.

Look at the Experimental Trajectory: both of these effects are evident from a comparison between target and sample temperature.