Allow Overlap - lets you place furniture overlapping with other furniture (e.g. placing a rug underneath an already placed table) - default true
Pickup If Under - lets you pickup rugs from underneath other furniture (this is the vanilla behaviour) - default false
You can change either of these options in GMCM or by editing the config file created after running the game.
This video shows the difference in behaviour:
Technical
Requires SMAPI, uses Harmony.
Implements a Generic Mod Config Menu interface to change config settings in-game.
Compatible with Mod Updater for automatic updates.
Code is at https://github.com/aedenthorn/StardewValleyMods.
If you want to complain or ask for help or help me test my mods, you can visit my Discord server.
A list of all my mods for Stardew Valley is available at https://www.nexusmods.com/stardewvalley/articles/895.