Most of the alt-photo process can be influenced by the paper selected. It is a good idea to use the paper intended for your final results. Papers, even of the highest quality, can vary from batch to batch. It is a good idea that once you find a paper you like, get a lot from that same batch. Yes, Edward Weston had said that a good print could even be made on a bath mat, but there are only a limited number of images that might look good that way.
Some thin papers can deliver unmatched sharpness, detail, and translucency.
Some thick papers can deliver great tooth, flatness, and texture. It is a mater of personal selection. Do not be surprised to find some papers costing more than commercial gelatin silver paper. Art stores are a good place to find several different papers. Once you find a paper you like, you may ask the store for a discount on purchasing a large quantity.