"A rising tide lifts all boats equally." Since bandwidth is measured in relation to the signal's peak amplitude, linear amplification alone should not increase the signal's bandwidth. Generally speaking, all frequencies produced by your source that fall in the amplifier bandwidth will increase by at least the gain of the amplifier. The amplifier's inherent nonlinearity will distort the input to some degree, which may increase some noise components more than others and even produce new noise components that will increase the signal bandwidth. Of course, operating the amplifier outside its specified limits will produce undesired products like "splatter."
We must be careful with our terms when talking about signals and bandwidth. The generator and the detector each has its own "bandwidth."
In theory, a "700Hz tone" has zero bandwidth. In reality, all processes are subject to random fluctuations and nonlinear behaviors that increase the bandwidth from this ideal. Inasmuch as an oscillator is an (imperfect) amplifier with positive feedback, noises that appear in the output are fed back to the input, resulting in self-mixing products that appear as noise in a spectrogram. (In fact, the "noise" of random fluctuations is what gets the oscillation process going in the first place!) Even a crystal oscillator produces "noise sidebands" that generally diminish with frequency difference from the carrier. The "bandwidth" of this real world "pure tone" depends on many factors including: component selection and construction; circuit design and implementation; environmental noises like vibration and power supply variations.
The detector measures the power that is present in its bandwidth. A wider detector will capture more of the generator's noise power than will a narrowband detector. Combining the properties of the generator with the properties of the detector gives meaning to your statement, re: "700Hz tone in a 3kHz bandwidth" but doesn't relate to the signal's bandwidth as a function of its amplitude.