Theoretical formula for the start-up time are as described in Start-up time (1)-Overview-

T_str: Start-up time
k : Proportionality constant
L₁ :Equivalent motional inductance of crystal unit
|-R|: Negative resistance of circuit (absolute value)
R_L: Load resonance resistance of crystal unit

The way to improve the startup time (to a smaller value) by changing the circuit is to "increase the negative resistance (absolute value) of the circuit".

The way to increase the negative resistance (absolute value) of a circuit by changing the circuit constants is described in What is "Negative Resistance"(3).
1) Reduce the capacitance of the external capacitors (Cin, Cout)
2) Reduce the value of damping resistance (Rd)
Therefore, implementation of 1) or 2) (or both) improves the startup time.

(*1) Feedback Resistance: properly feeds back current and signal from the output terminal to the input terminal of the IC
(*2) Damping Resistance: suppresses oscillation amplitudes