diff --git a/src/gmt_plot.c b/src/gmt_plot.c
index 3825e23ec5d..fe2c8fe103f 100644
--- a/src/gmt_plot.c
+++ b/src/gmt_plot.c
@@ -6155,6 +6155,11 @@ GMT_LOCAL void gmtplot_get_outside_point_extension (struct GMT_CTRL *GMT, double
 		L = 0.0;
 	else	/* Safe to get width (but still truncate to width of map) */
 		L = MIN (fabs (W / tan_angle), GMT->current.map.half_width);	/* L is positive */
+	/* For lines that cross the border at a shallow (near-tangent) angle, L can blow up far beyond
+	 * the segment that defines the crossing direction, producing a visible stray line segment
+	 * (https://github.com/GenericMappingTools/gmt/issues/5954). The extension should never need
+	 * to reach further than the segment itself, so cap it at that segment's length. */
+	L = MIN (L, hypot (dx, dy));
 	/* Compute the coordinates of the point at a distance L away from clip point in the direction of angle */
 	*x_off = x_on - L * cosd (angle);
 	*y_off = y_on - L * sind (angle);
diff --git a/src/seis/pssac.c b/src/seis/pssac.c
index 67394d45431..fcc6e895fbf 100644
--- a/src/seis/pssac.c
+++ b/src/seis/pssac.c
@@ -674,6 +674,8 @@ EXTERN_MSC int GMT_pssac (void *V_API, int mode, void *args) {	/* High-level fun
 	}
 	GMT_Report (API, GMT_MSG_INFORMATION, "Collecting %ld SAC files to plot.\n", n_files);
 
+	gmt_map_clip_on (GMT, GMT->session.no_rgb, 3);	/* Clip to map so border-crossing traces don't leak stray pen segments outside the frame */
+
 	for (n = 0; n < (int)n_files; n++) {  /* Loop over all SAC files */
  		GMT_Report (API, GMT_MSG_INFORMATION, "Plotting SAC file %d: %s\n", n, L[n].file);
 
@@ -937,6 +939,8 @@ EXTERN_MSC int GMT_pssac (void *V_API, int mode, void *args) {	/* High-level fun
 	gmt_M_free (GMT, L);
 	gmt_M_free (GMT, Ctrl->In.file);
 
+	gmt_map_clip_off (GMT);
+
 	if (Ctrl->D.active) PSL_setorigin (PSL, -Ctrl->D.dx, -Ctrl->D.dy, 0.0, PSL_FWD);	/* Reset shift */
 
 	PSL_setdash (PSL, NULL, 0);