What is the filter with the less phase shift?
Technically, the most reactive filter is the all-pass filter with a gain of 1, this filter has no phase shift at all. But it is not a really useful filter.

Here's what you need to take in account : 

1 - IIR versur FIR

An IIR filter will have less phase shift than a FIR filter for the same cut-off frequency

2 - Cut-off frequency

For an IIR low-pass filter,  a filter with 0.1 Fs cut-off frequency will have a higher phase-shift than a  with a 0.2 Fs cut-off frequency. An all-pass filter i.e. a gain of 1 has no phase shift.

3 - IIR filter order

For an IIR, the higher the order, the higher the phase shift.

4  - FIR number of taps

Assuming you a linear-phase low-pass FIR filter, the more taps you have the higher the phase shift. If you want a high cut-off frequency you can get by with less taps. If you want a really low cut-off frequency you will need more taps thus more phase shift.

5 - Minimum phase-filter

There is a class of filter called minimum-phase filter. They are the most reactive (fastest decay) for a given cut-off frequency according to this reference

https://www.dsprelated.com/freebooks/filters/Minimum_Phase_Filters.html