In theory, a rocket could "follow a rhumb line" starting with any kind of directional information. But there seem to be two use cases:
The very-early Pegasus rockets used a directional gyro derived from an aircraft gyrocompass, perhaps due to their aircraft parentage. This was later updated to a full IMU, eventually augmented to GPS.
Another approach is ...
“Sailing a rhumb line” means holding a constant compass bearing. For short distances, this stays close to a great circle path.But at longer distances and/or higher inclinations, the rhumb-line path “tends north” of a great circle as shown in the Questions globular image.
For a fast, short launch, a rhumb-line trajectory has the advantage of simplicity. As ...
The rovers used a gyroscope-based navigation system.
The Lunar Roving Vehicle (LRV) navigation system consists of a
directional gyro, a set of incremental odometers, and a hybrid
analog-digital signal processor plus appropriate controls and
Info from Lunar Roving Vehicle Navigation System Performance Review, (NASA Tech Note D-7469)
The Lunar Roving Vehicle did have a (form of) compass. It was gyroscopic rather than magnetic, thus it needed calibration when first powered up using the sun angle as a reference. It's in the upper left of the console here:
According to Wikipedia:
Navigation was based on continuously recording direction and distance through use of a directional gyro and ...