stochastic processes - Estimation of the drift of a non-stationary process

I'd like to estimate the drift of a continuous-paths, non-stationary, stochastic process $X_t$ from a time series of values $\{X_{i\Delta t}\}_{i=1,\dots,N}$ sampled from a single realisation of that process over $t \in [0,T]$.

Although the objective is to calibrate a pricing model (specified under $\Bbb{Q}$) based on historical series (observed under $\Bbb{P})$, we can forget about this and assume we're working under a single measure $\Bbb{P}$.

I've bumped into a similar question, but unfortunately the references provided apply to processes admitting a stationary distribution (= linear drift Ornstein-Uhlenbeck processes used for interest rates modelling) an assumption which I'd like to move away from.

Actually, my assumptions could even be further simplified to the case of a simple Arithmetic Brownian motion with drift

$$dX_t = \mu dt + \sigma dW_t$$

My question is: can someone point me towards a "nice" method to obtain an estimator $\hat{\mu}$ of $\mu$ from the observation of a $N$-sample $\{X_i\}_{i=1,...,N}$, where by "nice" I mean that the finite sample properties of the latter estimator should be better than the usual MLE (= LSE) estimator

$$\hat{\mu} = \frac{1}{N-1} \sum_{i=1}^{N-1} \Delta X_i$$ whose relative error is proportional to $1/\sqrt{N\Delta t}$ i.e. to the time horizon $T=N \delta t$ over which the data sample is collected but not to the number of data points $N$ itself making it almost useless in practice.

I'm particularly interested in answers where the answerer has had a successful experience in implementing the method he/she proposes in practice. Because I've already come across different approaches/algorithms myself, but none were satisfying as far as I am concerned.