What does one do when the coefficient for the log of the rate estimator for a poisson rate model is very different from 1?

Let's say we have "number of accidents" as the response in a poisson regression model. One of the predictors is "number of days." Naturally, we expect more accidents to occur over more days, so it makes sense to treat this as a rate model and consider "number of accidents per day."

So we take $$\log \frac{accidents}{days} = \log(accidents) - \log(days)$$ and end up with a model like $$\log(accidents) \sim \log(days)\ + \ldots + (other\ predictors)$$

Typically we would expect the coefficient on log(days) to be close to 1, at least according to my reading thus far: We expect the occurrence of interest to be proportional to the amount of time. As such, we would "offset" log(days) in our stats software, forcing the coefficient to be 1.

Suppose, however, we don't offset "days," and the parameter ends up being something nowhere near 1. What does that mean (if anything)? Does this suggest that it is inappropriate to use the offset because there isn't a constant rate of events of interest per time period?

Say, for instance, our statistics software tells us the coefficient for log(days) is 3. That suggests a multiplicative effect of $e^3(days)$ on number of accidents, if I understand Poisson regression coefficients correctly. Does this mean it would be inappropriate to treat "days" as a component of a rate? Would we instead need to investigate some other time-related component that could impact the response? Or would we offset "days" with a coefficient different from 1?

Thank you!