Notch filtering line noise outside of frequency band of interest
A general rule of thumb is to filter as little as necessary, because every filter distorts your desired signal, even if just a little bit. Also: a notch filter is advised in a situation where narrow-band interference is present within your signal's band. The notch filter will reject the interference while having minimal effect (ideally) in your signal.

In the problem you describe, my preferred approach would be to go with a low-pass filter only, and increase its order to achieve sufficient rejection of the 60 Hz noise.