Estimating Low Frequencies (including amplitude and phase)
In the frequency representation given by the FFT there are no frequencies missing and their is no binning of frequencies.

FFT decomposes a discrete signal in to a sum of sinusoids of certain frequency and phase. This representation is exact and has the same energy as the original signal. No information is lost, as you can do an inverse FFT and recover the original signal exactly. The binning is coming from the discretisation of the signal, not the FFT.

Also for a finite length signal, there are no frequencies lower than the one with the period equal to the length of the signal. This is from the mathematical definition of the spectrum. Put more simply, the DFT/FFT 'assumes' the signal is repeating. Zero padding of the signal therefore increases the period and therefore the lowers the first frequency.

If there are lower frequencies due to some prior knowledge, e.g. if we sampled a sine wave over half its period, then this information must be included.

One solution may be to roll your own spectrum and scale the frequency components with period longer than that of the signal. The assumption being that the low frequency component continues.

For example, let $f_S$ be the frequency you are interested in, with period = $1/f_s &gt; 1$ longer than the period of the signal which is normalised to 1.

Now you convolve the signal with the sine and cosine waves at that frequency. No zero padding.

$$f_e = \sum_t^N f[t] \times cos[2\pi f_s t]$$
$$f_o = \sum_t^N f[t] \times sin[2\pi f_s t]$$

Then multiply by $1/ f_s$ to account for the assumption the signal outside of the sample continues like this, and multiply by 1/N to account for number of samples.

$$f_e' = f_e \times 1/ f_s \times 1/N$$
$$f_o' = f_o \times 1/ f_s \times 1/N$$

Amplitude and phase are then given in the usual way.

This can be done at whatever frequency graduations you desire.

Edit:
Changes with regard to first comment.

Edit 2:
Answer riddled with errors. Updated.

Edit 3: Add factoring for N

--
Edits welcome if I have this wrong.