I'm running a single tool--Zonal Statistics by Table--that should overwrite the output as a table. I'm not using the default name given by the tool.
The workaround I've developed is to write the zonal stats output to a file gdb, then run Table to Table to transfer the table from the file gdb to the personal gdb (the only reason I'm using the personal gdb is that it has numerous forms and reports within it that work with these tables). Looks like it might be something specific to the Zonal Stats tool, as I have no problem with Table to Table overwriting the output.
Thanks for the suggestion, though.