i think you should use WCS instead of WMS for sharing raster datasets on the Web. AFAIK, wms dont have this capabilities for sharing tiff images.
Unlike OGC Web Map Service (WMS), which portrays spatial data to
return static maps (rendered as pictures by the server), the Web
Coverage Service provides available data together with their detailed
descriptions; defines a rich syntax for requests against these data;
and returns data with its original semantics (instead of pictures)
which may be interpreted, extrapolated, etc., and not just portrayed.
Unlike OGC Web Feature Service (WFS), which returns discrete
geospatial features, the Web Coverage Service returns coverages
representing space/time-varying phenomena that relate a
spatio-temporal domain to a (possibly multidimensional) range of
properties. As such, WCS focuses on coverages as a specialized class
of features and, correspondingly, defines streamlined functionality.
and you can check out WCS info for geoserver here.
Benefits of WCS
WCS provides a standard interface for how to request the raster source
of a geospatial image. While a WMS can return an image it is generally
only useful as an image.The results of a WCS can be used for complex modeling and analysis, as it often contains more information. It also allows more complex querying - clients can extract just the portion of the coverage that they need.
WCS output formats
GeoTiff - (format=geotiff)
GTopo30 - (format=gtopo30)
ArcGrid - (format=ArcGrid)
GZipped ArcGrid - (format=ArcGrid-GZIP)
i hope it helps you..