I hope that my recent paper is helpful for the discussion: Choi et al. (2023),  https://arxiv.org/abs/2302.08758 .

The paper finds quite many new IV bounds (both upper and lower) for a given price (this is main theme of the paper).
Then, it also formulates a new Newton-Raphon method on the log price to handle very small option premiums properly. The iteration formula is quite simple (see Eq. (30) in the paper), and it always converges as long as the initial guess is a lower bound. (The proof is a bit tedious. Just see the left side of Figure 1. The log price is a concave function of sigma.) 
So we use a lower bound found in the earlier part of the paper (specifically L3 in Eq (23)). 

I actually tested the famous example in this thread (S0=1, K=1.5, sigma=4%) and reported it in the paper (See Section 3.3 Numerical Example(. The new NR method reaches an IV value within 2e-11 just after three iterations. 

This looks like NR with some upfront work to find an initial seed (which is always the main challenge/bottleneck).  
Mention is made of iterating a number of times to get a desired accuracy which may affect performance.