Assuming you use Stata (I base that assumption on your use of terminology) you can use your second model. It is important that the final iteration does not show such messages, but such messages in the middle of the iteration log are not a problem and not surprising for a zero-inflated negative binomial (zinb) model.
One thing you can do is make sure that the value 0 on all explanatory/independent/right-hand-side/x-variables are within or near the range of the data. For example, in most case you don't want to use year of birth, as the year 0 is usually way outside the range of your data. Instead you would create and use a different variable containing something like (year of birth - 1960). Also you want to scale your variables such that their effects are of the same order of magnitude. For example don't add annual income in euros, as a single euro increase in annual income isn't going to do much. Instead add annual income in 1000s of euros. Similarly, I usually enter year of birth measured in decades since 1960. Most of the time I do both because it makes the coefficients easier to interpret, but in complicated models like zinb it can actually help attain convergence.