I noticed the Extract Values to Points allows the user to extract all attributes within a raster dataset. However, I do not see that capability within Extract Multi Values to Points. Am I missing something?
Extract Multi Values to Points—Help | ArcGIS for Desktop
You can extract band information if it is an image
You can extract information from multiple rasters
But... any tabular information...which I presume you have joined, will not be extracted.
You might be able to join the extra tabular information that you need, since the tool, by design, extracts the cell value from the raster and a cell can only have one 'value'
Integer rasters can have additional attributes and as Dan suggests you can join these raster tables to the output point feature class then use CopyFeatures to persist the join.
Retrieving data ...