Linear Chirp parameters from a segment of the signal Edit:" please see the attached figure. The blue part is all I have. Also, you can neglect the noise term. Assume the signal is deterministic. Typical values of the unknown parameters $\alpha$ and $\omega_0$ are in the the range [0.25,5]. Note that all the parameters are unknown $A \omega_0, \alpha, \phi_0$. The setup hints at curve fitting which I did but don't prefer."

I have a small part from a chirp signal

$$s(t)= A \cos\left(\omega_0 t+ \frac\alpha2 t^2 + \phi_0\right)+n(t)$$

with very low start frequency $\omega_0$  and chirp rate $\alpha$.

The signal is sampled at high sampling frequency $f_s$. But the available time domain signal is short, i.e. barely a complete cycle. The amplitude of the signal and the phase shift are also unknown. Are there ways to figure out the chirp parameters other than curve fitting or spectrogram?
Amplitude $A$ is trivial. Square and low-pass filter/average; then take the square root.

Without noise (or with good SNR), you could simply calculate a derivative:

\begin{align}\frac{\partial\, s\}{\partial t} = -A\sin\left(\omega_0t + \frac\alpha2t^2 +\phi_0\right)\cdot\left(\omega_o+ \alpha t\right)
\end{align}

and low-pass filter the result

$$\text{LPF}\left\{\frac{\partial\, s\}{\partial t}\right\} = \omega_o + \alpha t$$

From the resulting linear function, $\alpha$ can directly be extracted as the slope, and knowing the slope, $\omega_0$ is just the constant summand.

With $A$, $\omega_0$ and $\alpha$ known, you can simply compare at any fixed $t$ to find $\phi_0$.