Say you have a function $f(t)=B\cdot e^{\alpha t}$ which is an LSM-estimation of some measurement data and that you have reason to believe that the data in a long term better would be modelled as a logistic function $g$: $$g(t)= \frac{L}{1+e^{-\beta\cdot(t-t_0)}}$$ where $L$ is the upper limit of $g$ and $g(t_0)=\frac{L}{2}$.
Is it possible estimate a smallest value $t_0$ (and $L$) from the collected data, especially from the expected errors of the parameters $B$ and $\alpha$?
Some context:
The Keeling Curve is a graph of the accumulation of carbon dioxide in the Earth's atmosphere based on measurements taken at the Mauna Loa Observatory. The annual mean of measured data $(K)$ minus a pre-industrial level $P=258.6$ ppm is approximated with
$4.243\cdot 10^{-13}\cdot e^{0.0166\cdot t}$ using LSM with $r^2>0.999$.
Year K(ppm) Err
----------------
1959 315.98 0.12
1960 316.91 0.12
1961 317.64 0.12
1962 318.45 0.12
1963 318.99 0.12
1964 319.62 0.12
1965 320.04 0.12
1966 321.37 0.12
1967 322.18 0.12
1968 323.05 0.12
1969 324.62 0.12
1970 325.68 0.12
1971 326.32 0.12
1972 327.46 0.12
1973 329.68 0.12
1974 330.19 0.12
1975 331.12 0.12
1976 332.03 0.12
1977 333.84 0.12
1978 335.41 0.12
1979 336.84 0.12
1980 338.76 0.12
1981 340.12 0.12
1982 341.48 0.12
1983 343.15 0.12
1984 344.85 0.12
1985 346.35 0.12
1986 347.61 0.12
1987 349.31 0.12
1988 351.69 0.12
1989 353.20 0.12
1990 354.45 0.12
1991 355.70 0.12
1992 356.54 0.12
1993 357.21 0.12
1994 358.96 0.12
1995 360.97 0.12
1996 362.74 0.12
1997 363.88 0.12
1998 366.84 0.12
1999 368.54 0.12
2000 369.71 0.12
2001 371.32 0.12
2002 373.45 0.12
2003 375.98 0.12
2004 377.70 0.12
2005 379.98 0.12
2006 382.09 0.12
2007 384.03 0.12
2008 385.83 0.12
2009 387.64 0.12
2010 390.10 0.12
2011 391.85 0.12
2012 394.06 0.12
2013 396.74 0.12
2014 398.87 0.12
2015 401.01 0.12
2016 404.41 0.12
2017 406.76 0.12
2018 408.72 0.12
2019 411.66 0.12
2020 414.24 0.12
See also https://economics.stackexchange.com/questions/47480/is-climate-change-an-economic-question
In the question the model is $$\boxed{f(t)=A+B\:e^{\alpha\:t}}\quad \begin{cases} A=258.6\\ B=4.243\:10^{-13}\\ \alpha=0.0166 \end{cases}\quad\to\quad \text{RLMSE}=0.732$$ From the given data and with non-linear regression for least mean square error I found this : $$\begin{cases} A=256.001\\ B=9.635494\:10^{-13}\\ \alpha=0.016203 \end{cases}\quad\to\quad \text{RLMSE}=0.698$$ Both are consistant. On the next figure one cannot see a difference beween the two corresponding curves.
It should be surprising that a logistic model would improve the fitting compared to the exponential model because the shape of the curve is not in the range where a ceiling begins to appear. The data is outside the range where a logistic function can be well determined.
With the logistic model $$\boxed{g(t)= \frac{L}{1+e^{-\beta\cdot(t-t_0)}}}$$ the non-linear regression fails to accurately determine $t_0$. For any different couples $(L$ , $t_0)$ the fitting is quite the same. For example, for the same RLMSE than above we get :
$$\begin{cases} A=256.107\\ L=84758.206\\ \beta=0.01624\\ t_0=2407 \end{cases}\quad\to\quad \text{RLMSE}=0.698$$
Comparing the exponenial model and the logistic model, it is easy to show the relationships : $$\beta \simeq \alpha$$ $$L\simeq B\:e^{\alpha\:t_0}\quad\implies\quad t_0\simeq \frac{1}{\alpha}\ln\left(\frac{L}{B}\right)$$ which are valid in the range $t<<t_0$.
$t_0$ is far outside the range of the experimental data. All this confirmes that the estimates values of $L$ and $t_0$ are nearly without signifiance.