The use of cone-beam computed tomography (CBCT) in image-guided radiation therapy (IGRT) has helped linear accelerators (LINACs) become the most popular form of radiation therapy today. The ability to acquire tomographic information from the patient at the treatment position allows for setup and target verification as well as steeper dose gradients and higher dose fractions while simultaneously providing images that allow the oncologist to monitor the tumor's response to the therapy. The current kV-CBCT scanning configuration of this LINAC-mounted imaging system provides a circular trajectory of the source and detector around the patient as the LINAC gantry makes a single rotation around the patient. Though this provides the requisite trajectory for the analytic-base FDK reconstruction algorithm that is the workhorse of clinical reconstruction in IGRT today, there are some issues of this scanning geometry that can either limit or even prevent the use of this CBCT information in clinical practice. In this work, we develop a generalized non-circular scanning trajectory framework enabled by optimization-based reconstruction that allow for non-circular trajectories that directly address two issues of LINAC-mounted CBCT for IGRT. The first issue is overcoming the limited axial coverage provided by the current detector size and circular acquisition trajectory. This is problematic as the CBCT axial coverage is smaller than the potential treatment field size. As engineering costs restrict the axial coverage of the detector, we investigate potential non-circular trajectories that can extend the axial coverage with current LINAC-mounted CBCT detectors. The other existing limitation that could be resolved with non-circular scanning trajectories is that of potential patient collisions with the LINAC gantry. As some patient treatment positions can put the patient in a collision path with components of the LINAC gantry, the inability to acquire the full circular rotation can lead to forgoing the CBCT. This is another problem to which we provide example trajectories that could alleviate these collisions while still acquiring useful CBCT images. We found that in both of these examples, our non-circular imaging framework was able to reconstruct images that have comparable image quality to the current clinical method using a single circular scan while simultaneously providing potential solutions to the current clinical limitation of restricted axial coverage and potential patient collisions with the LINAC gantry.