How can i achieve texturing of a dynamic generated Hex?

I have dynamically created a hex and i am trying to add a texture/material to it but it does not look good.

This is the code i am using:

 List<Vector3> vertices;
     List<int> triangles;
     public float floorLevel = 0;
     public Material material;
 public void GenerateMesh()
     {
         MeshFilter meshFilter = gameObject.AddComponent<MeshFilter>();
         Renderer renderer = gameObject.AddComponent<MeshRenderer>();
 
         vertices = new List<Vector3>();
         triangles = new List<int>();
 
         Hex hex = new Hex(
             new Node(new Vector3(-1, floorLevel, -.5f)),
             new Node(new Vector3(-1, floorLevel, .5f)),
             new Node(new Vector3(0, floorLevel, 1)),
             new Node(new Vector3(1, floorLevel, .5f)),
             new Node(new Vector3(1, floorLevel, -.5f)),
             new Node(new Vector3(0, floorLevel, -1))
             );
 
         Vector2[] uv = new Vector2[]
         {
             new Vector2(0, .25f),
             new Vector2(0, .75f),
             new Vector2(.5f, 1),
             new Vector2(1, .75f),
             new Vector2(1, .25f),
             new Vector2(.5f, 0)
         };
 
         TriangulateHex(hex);
 
         Mesh mesh = new Mesh();
         mesh.vertices = vertices.ToArray();
         mesh.triangles = triangles.ToArray();
         mesh.uv = uv;
         mesh.RecalculateNormals();
 
         meshFilter.mesh = mesh;
         meshFilter.transform.rotation = Quaternion.Euler(0, 90f, 0);
 
         renderer.material = material;
     }

These are the images:

I want the hexagon texture to tile perfectly on the generated hexagon, but it tiles weirdly. Does anyone know what the problem is?

Thanks.

Later edit: Based on the comments bellow i tried to remake this hex, but still haven't managed to get it working

Here is a representation of what i did:

So based on this image i created the following vertex positions:

• left: 0, .5f

• top-left: .25f, 1

• bottom-left: .25f, 0

• top-right: .75f, 1

• bottom-right: .75f, 0

• right: 1, .5f

I have created the following triangles based on corner positions:

• 1, 2, 6

• 2, 5, 6

• 2, 3, 5

• 3, 4, 5

The hex renders ok.

I then try to add the UV mapping on the hex. These are the points i use for the mapping (same vertex positions as above):

• 0, .5f

• .25f, 1

• .75f, 1

• 1, .5f

• .75f, 0

• .25f, 0

Still the texture does not tile over the generated hex, it is skewed like in the image above, in a different way, but still not correctly tiled. What am i doing wrong?