1. When you are using not-so-easily-converging method for convergence :
Using simplest case of ATMF option , we can assume that
S = X * exp(-rT)
This gives a closed form solution for implied volatility as follows, which can be safely used as your initial estimator
ImpliedVol = (C/S) * sqrt ( 2*pi / T)
C = Market price of call option
S = Underlying asset price
Reference :- This was presented by Brenner and Subrahmanyam (1988) in one of the papers.
2. When you're using a better converging method , like Newton's or the similar league
Calculate your historical vol , and then the option price. Then using linear interpolation, scale up or down your historical vol w.r.t ratio/difference between your your option price ( BS ) and the market price
IV = histVol * (C / Cbs) * (Optional constant by intuition , default = 1.0)
C = Market price
Cbs = black scholes price of option
I would say that's a quite rough estimate and won't be very close to the actual value. But since you're using Newton Ralphson's approach which is quite good at convergence, I'd say this could be pretty smart one for an initial guess. Please feel free to use your own value for 'optional constant' here.
Reference : My thoughts.