New Mesh with Existing Curve, UV coordinates, Curve Modifier
The cylinder mesh can be UV Mapped in the standard fashion. The Curve modifier beforms the cylinder into a spiral in a second step. Shader Nodes allow the user to specify UV coordinates in nodes. This allows image placement in UV directions and animation of location. This flexibility may be important because the original question has spiral and the long cylinder. Blender has many options for manipulating UV Coordinates.
Please consider the following steps. The steps are abbreviated and can be more detailed according to comments.
- Use your existing spiral
- Add cylinder mesh. Mesh length will probably much longer than diameter. In edit mode add seams. UV Unwrap.
- Align mesh origin and curve origin. Add Curve Modifier to mesh and adjust to suit your tastes. You many need to adjust cylinder length and position and rotation about long dimension.
Image Above. High Level. A Spiral Curve exists similar to yours. A Separate UV Mapped Cylinder mesh with UV Seams. Mesh was UV Mapped in its original state in edit mode. The Cylinder was quite long to occupy the Curve. The circumference is much smaller than length. Please see the Curve Modifier which allows mesh to follow spiral. Curve Modifiers works best when the origin of mesh and curve coincide. Please see other educational content on Curve Modifier if that suits you. Zoomed out view to show some texture on a restricted part of mesh. Zoomed in view to show texture detail.
Image above. Shader Nodes. Please inspect and improve. Other approaches are possible. Upper Right Frame designates the UV coordinates of the texture to be placed ... and remaps to [0,1]. Note the reader uses the numerical coordinates to place the final image. Thus you can slide the image around the length and around the shorter diameter. An image derived from OP question image was used.
Image Above. Cylinder in edit mode for UV Unwrap respecting seams. Seams are visible and rather difficult to see in the image. Partial selection of faces and thus partial appearance in UV editor area.
Please Click to zoom.