Given a real signal, increase modulation frequency
You can simply multiply by another cosine: $x(t)\cos(2\pi f_ct)\cos(2\pi 0.001f_ct)=x(t)\cos(2\pi0.999f_ct)+x(t)\cos(2\pi1.001f_ct)$ (ignoring scale factors). If $x(t)$ is very narrow in frequency, so that the spectra of the two terms above don't overlap, you can use a high-pass filter to do what you want.

If $x(t)$ is not narrowband enough, you can shift $x(t)\cos(2\pi f_ct)$ two times. With adequate filtering, you will obtain the desired shift.

Simply multiplying by a complex exponential is not going to work, since it will shift the entire spectrum of $x(t)$ up or down, and the resulting signal will be complex. You want to shift the positive frequencies of $x(t)$ "to the right", and its negative frequencies "to the left".

However, you can use the complex exponential in the following way: Filter out all the negative frequencies of the modulated signal (using for instance the Hilbert transform), and up-convert them using a complex exponential. Then, filter out all the positive frequencies, and down-convert the result. Then, add the down-converted and up-converted signals.

So, your options are:


Use one or two cosine signals and a few filters (all real).
Use two complex filters and two complex exponentials.