According to energy conservation principle,

If, $x _1$ is maximum elongation in the spring when the particle is in its lowest extreme position. Then,

$mgh =\frac{1}{2} kx _1^2- mgx _1$

$\Rightarrow \frac{1}{2} kx _1^2- mgx _1- mgh =0$

$\text { or, } x _1^2-\frac{2 mg }{ k } x _1-\frac{2 mg }{ k } \cdot h =0$

$\therefore x _1=\frac{2 mg }{ k } \pm \sqrt{\left[\left(\frac{2 mg }{ k }\right)^2+4 \times \frac{2 mg }{ k } h \right]}$

Amplitude $A= x _1 -x _0$ (elongation in spring for equilibrium position)

$A =\frac{ mg }{ k } \sqrt{\left(1+\frac{2 hk }{ mg }\right)}$