I am looking for a bound on the principal branch of Lamber W-function $W(x)$ that works well when $x$ is approaching $-\frac{1}{e}$.
There are several bounds like this bound \begin{align} W_{0}(x)\leq \ln x-\ln \ln x+{\frac {e}{e-1}}{\frac {\ln \ln x}{\ln x}} \end{align} which holds for $x> e$.
On
Doesn't work very well as asymptotic expansions about $x=-e^{-1}$, but if all you need is bounds, then
$$f_n(x;a)=\begin{cases}ax,&n=1\\x\exp(-f_{n-1}(x)),&n>1\end{cases}$$
give the simple bounds
$$f_n(x;e)\le W_0(x)\le f_n(x;1)\tag{$-e^{-1}\le x\le0$}$$
for all $n$. These work best around $x=0$ though...
Not bounds but approximations.
Around $x=-\frac 1 e$, we can build quite accurate approximations of $W(x)$ using $[n,n]$ Padé approximants such as
$$W(x) \sim \frac{-1+\frac{14 }{45}t+\frac{301 }{1080}t^2 } {1+\frac{31 }{45}t+\frac{83 }{1080}t^2} \tag1$$ $$W(x) \sim \frac{-1-\frac{974 }{22659}t+\frac{21865 }{51792}t^2+\frac{89131 }{1165320}t^3 } {1+\frac{23633 }{22659}t+\frac{104225 }{362544}t^2+\frac{3167 }{196560}t^3 } \tag2$$ $$W(x) \sim \frac{-1-\frac{11637254 }{29330279}t+\frac{463636649 }{1055890044}t^2+\frac{23930361857 }{110868454620} t^3+\frac{192684057311 }{10643371643520}t^4} {1+\frac{40967533 }{29330279}t+\frac{659231191 }{1055890044}t^2+\frac{1928737771 }{20157900840} t^3+\frac{34384971553 }{10643371643520}t^4}\tag3$$ where $\color{red}{t=\sqrt{2(1+e x)}}$.
Computing for for a few values of $x$ $$\left( \begin{array}{ccccc} x & (1) & (2) & (3) & \text{exact} \\ -0.36 & -0.806085 & -0.806084 & -0.806084 & -0.806084 \\ -0.35 & -0.716641 & -0.716639 & -0.716639 & -0.716639 \\ -0.34 & -0.653701 & -0.653695 & -0.653695 & -0.653695 \\ -0.33 & -0.603280 & -0.603267 & -0.603267 & -0.603267 \\ -0.32 & -0.560511 & -0.560490 & -0.560489 & -0.560489 \\ -0.31 & -0.523023 & -0.522990 & -0.522990 & -0.522990 \\ -0.30 & -0.489448 & -0.489403 & -0.489402 & -0.489402 \\ -0.29 & -0.458918 & -0.458857 & -0.458856 & -0.458856 \\ -0.28 & -0.430837 & -0.430760 & -0.430759 & -0.430759 \\ -0.27 & -0.404781 & -0.404685 & -0.404684 & -0.404684 \\ -0.26 & -0.380432 & -0.380314 & -0.380313 & -0.380313 \\ -0.25 & -0.357545 & -0.357405 & -0.357403 & -0.357403 \\ -0.24 & -0.335929 & -0.335763 & -0.335761 & -0.335761 \\ -0.23 & -0.315428 & -0.315236 & -0.315233 & -0.315233 \\ -0.22 & -0.295916 & -0.295696 & -0.295693 & -0.295692 \\ -0.21 & -0.277288 & -0.277039 & -0.277035 & -0.277034 \\ -0.20 & -0.259457 & -0.259176 & -0.259171 & -0.259171 \\ -0.19 & -0.242347 & -0.242033 & -0.242028 & -0.242028 \\ -0.18 & -0.225895 & -0.225546 & -0.225540 & -0.225540 \\ -0.17 & -0.210045 & -0.209660 & -0.209653 & -0.209652 \\ -0.16 & -0.194748 & -0.194326 & -0.194317 & -0.194317 \\ -0.15 & -0.179962 & -0.179501 & -0.179491 & -0.179491 \\ -0.14 & -0.165650 & -0.165149 & -0.165138 & -0.165138 \\ -0.13 & -0.151779 & -0.151236 & -0.151224 & -0.151223 \\ -0.12 & -0.138318 & -0.137732 & -0.137719 & -0.137718 \\ -0.11 & -0.125241 & -0.124611 & -0.124596 & -0.124596 \\ -0.10 & -0.112525 & -0.111849 & -0.111833 & -0.111833 \end{array} \right)$$
If you want simple bounds, using series expansion, you have $$W(x) < -1+t \qquad \text{and} \qquad W(x) >-1+t-\frac 13 t^2$$