https://time.com/6116644/earth-second-moon/

There's a rock called Kamo’oalewa in a corkscrew orbit around the moon and earth for the next 300 years anyway, at 40-100x the distance of the moon from Earth, so pretty distant, only identified in 2016. Kamo'oalewa is around the size of a school bus, so not that big. Its orbital period is 366 days, a bit slower than Earth's. https://www.spacereference.org/asteroid/469219-kamooalewa-2016-ho3

I love this Spacereference website for the listings and excellent orbital simulations, BTW. You can change the plane and angle. It's like Outer space Zillow, only less invasive of privacy and without bogus pricing.

Anyway, Kamo'oalewa has another claim to being a moon: They finally analyzed why this chap looked "off" and less shiny compared to other asteroids. It actually has the right composition and albedo to have been part of the Moon itself (Luna has lots of craters and some of its rock has been blasted to Earth before.)

... OTH, that other "second moon?" Cruithne 3753 is an asteroid, around the size of Mt. Everest, which is co-orbital with Earth. It has an skewed orbit which crosses Earth's orbit, but never gets close to us. From our vantage, it moves in a horseshoe as it orbits the sun, as its actual orbit time is almost exactly that of Earth's, at 364 days. It also rotates on its axis once every 27 hours (its "day.")

Its actual (highly tilted) orbit takes it almost as close to the sun as Mercury, and further out than Mars, so it's pretty eccentric. It will not make any close approaches to Earth; it's almost in perfect resonance. It is not considered viable for Earth exploration. It is just one of multiple asteroids classified as "Near Earth Asteroids" (Aten type.)

This website has excellent orbital stimulations at the bottom of the page. https://www.spacereference.org/asteroid/3753-cruithne-1986-to

NASA doesn't just rocket, it also rocks.