Can you increase frequency resolution of FFT without increasing window size?
If you really insist on using FFT (rather than parametric methods, which wouldn't suffer from time/frequency trade-offs), you can fake a much better resolution by using the phase information to recover the instantaneous frequency for each FFT bin. Partials can then be detected by looking for plateaus in the function giving instantaneous frequency as a function of FFT bin index. The common implementation of this technique as described in this paper will "cost" you one extra STFT (instantaneous frequency is recovered by operations on the STFT of the signal, and STFT of the derivative of the signal).

See for example the ifgram function in this Matlab implementation of sinusoidal modeling of audio signals.

Note that this won't help resolving two partials falling into adjacent FFT bins. It will just provide a much more accurate frequency estimate than just converting into a frequency the FFT bin index of a spectral peak.