Don’t put extension cords under rugs.
You shouldn’t. Even a flat extension cord isn’t safe under a rug—it can overheat or get damaged and start a fire. Plus it’s a tripping hazard if the rug shifts.
Flat cords are less bad than round ones because they’re thinner and don’t create as big a bump. But “less bad” isn’t “safe.” The National Electrical Code says no extension cords under rugs or carpet for a reason. The cord can’t dissipate heat, and foot traffic slowly breaks the insulation.
If you absolutely need power there, a better option is either a floor cord cover (a rubber channel that protects the cord and lies flat) or calling an electrician for a floor outlet. That’s real money but real safe.
Your rug doesn’t need to be a fire hazard. Find another path for the cord.