Not in a meaningful way. Converting an 8-bit image to 16-bit only changes how the existing values are stored; it does not recover missing tonal detail from the original scan.

Interpolation can create intermediate values when resizing, but that is tied to changing spatial resolution, not magically restoring original color precision. In theory, downsampling can trade spatial resolution for a little more tonal precision, but for normal photo use this is rarely worthwhile.

If your concern is banding/posterization, higher bit depth after the fact usually won’t fix it. The original 8-bit steps are already baked into the scan. A practical workaround is dithering or adding a small amount of noise/grain, which can make gradients appear smoother by breaking up visible bands.

So: yes, you can convert the file to a higher bit depth, but no, it generally won’t add real color information. If possible, the best solution is to rescan from the original at higher bit depth.