Although visual search appears largely random, several oculomotor biases exist such that the likelihoods of saccade directions and lengths depend on the previous scan path. Compared to the most recent fixations, the impact of the longer path history is more difficult to quantify. Using the step-selection framework commonly used in movement ecology, and analyzing data from 45-second viewings of "Where's Waldo?", we report a new memory-dependent effect that also varies significantly between individuals, which we term self-crossing avoidance. This is a tendency for saccades to avoid crossing those earlier in the scan path, and is most evident when both have small amplitudes. We show this by comparing real data to synthetic data generated from a memoryless approximation of the spatial statistics (i.e. a Markovian nonparametric model with a matching distribution of saccade lengths over time). Maximum likelihood fitting indicates that this effect is strongest when including the last $\approx 7$ seconds of a scan path. The effect size is comparable to well-known forms of history dependence such as inhibition of return. A parametric probabilistic model including a self-crossing penalty term was able to reproduce joint statistics of saccade lengths and self-crossings. We also quantified individual strategic differences, and their consistency over the six images viewed per participant, using mixed-effect regressions. Participants with a higher tendency to avoid crossings displayed smaller saccade lengths and shorter fixation durations on average, but did not display more horizontal, vertical, forward or reverse saccades. Together, these results indicate that the avoidance of crossings is a local orienting strategy that facilitates and complements inhibition of return, and hence exploration of visual scenes.