How to modify the z depth value inside the vertex function of a surface shader?

I wrote the following surface shader. The idea is to increase the Z depth value of vertices, after projection.

Shader "Custom/Highlighted"
{
    Properties
    {
        _Color ("Main Color", Color) = (1,1,1,1)
    }
    SubShader
    {
        Tags { "RenderType"="Opaque" }

        CGPROGRAM
        #pragma surface surf Lambert
        #pragma vertex vert 

        struct Input
        {
            float4 pos: SV_POSITION;
            float4 color : COLOR;           
        };
        
        void vert(inout appdata_full v, out Input o) 
        {
            UNITY_INITIALIZE_OUTPUT(Input, o);       //
            o.pos = mul(UNITY_MATRIX_MVP, v.vertex); // does not work
            o.pos.z += 0.001;                        //
            o.color = v.color;                       //
        }

        fixed4 _Color;
        void surf (Input IN, inout SurfaceOutput o)
        {
            o.Albedo = _Color.rgb;
            o.Alpha = _Color.a;
        }
        ENDCG
    }
    Fallback "Diffuse"
}

The shader compiles, and objects are rendered correctly, however it's like the z depth value is not modified at all (what is inside vert is ineffective).

Here is the equivalent code as a fragment shader (which work perfectly) :

struct vertInput {
    float4 pos : POSITION;
    float4 color : COLOR0;
    float4 uv : TEXCOORD1;
};

struct vertOutput {
    float4 pos : SV_POSITION;
    fixed4 color : COLOR0;
};

vertOutput vert (vertInput input)
{
    vertOutput o;
    o.pos = mul(UNITY_MATRIX_MVP, input.pos);
    o.pos.z += 0.001;
    o.color = input.color;
    return o;
}