2n2222a as a Switch

The output high voltage from the emitter of Q1 will never exceed Vb-Vbe. If the maximum base drive voltage is 5V, then the emitter will never be able to rise above approximately 4.3 volts when Q1 is saturated.

If the power supply increases to 5.9 Volts, then the transistor is operating in a linear region and the difference of 5.9-4.3 is dissipated across the CE junction. As soon as the emitter tries to rise up, it clamps off the base and shuts down a little until it reaches equilibrium. If the trace you show is of the actual circuit and not of a simulation, then the ringing is most likely cause by some noise source in the circuit as 1Hz is to slow for it to be a transient response.

The upper end of R1 never floats. It is bound by the Vbe junction of Q1.

If you wish to use a high side driver, then perhaps consider a PNP device. If you want a low side driver, then use an NPN device. this will free R1 from Vbe.

This is a commonly encountered gotcha in the analog world.

Emitter followers are prone to oscillation when there is a stiff (eg. bypassed) base source and some stray inductance kicking around, especially when they have a capacitive load (such as a scope probe). If you look at the arrangement, it's really a Colpitts oscillator!

Your oscillation looks to be relatively low frequency, but it's probably your digital scope lying to you (aliasing), as they are wont to do.

Try a series base resistor of 100 ~ 1000 ohms and I bet the oscillation disappears.