Your goal is to get a sensitive measure of the change in hydrogen ion concentration as you add acid. If you are close to the equivalence point in a single point titration, then you can better achieve this. An uncertainty of 0.001 in measuring the final pH is equivalent to an uncertainty of about 0.23% in the hydrogen ion concentration. If the pH is 4.0, this is a significantly smaller absolute number than if it is 3.5.
However, sensitivity is not your only problem! You also need to minimize bias. The largest contribution to bias is in your pH measuring system's calibration, which needs to be done ahead of time for a one-point method.
The next largest contributor to bias is bicarbonate ion that has not reacted. There are a variety of ways to deal with this. The best is to bubble the acidified solution with air for a few minutes so as to remove CO2, thus reducing total carbon, and hence the absolute amount of bicarbonate ion present at any particular pH. A slightly lower pH also reduces the proportion of the total carbon remaining that is present as bicarbonate.
Also, in a one-point method the amount of acid you add needs to ensure you get an OK alkalinity for all the samples you intend to measure. Unless you adjust it with some a priori knowledge as to the true alkalinity, you are bound to have compromises.
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