A bedrail clamp for attaching an accessory to a bedrail of a surgical/medical bed at a desired location without having to slide the bedrail clamp from an end of the bedrail until the accessory is in the desired location, the bedrail clamp comprising: (a) a clamp housing that includes (i) a throat area having a first bedrail surface and being configured to receive the bedrail and (ii) a first housing guide element; (b) an actuating member coupled to the clamp housing and configured to move relative to the clamp housing between a clamp-open position and a clamp-closed position, wherein, when the bedrail is in the clamp housing's throat area, the actuating member is nearer to the bedrail in the clamp-open position than in the clamp-closed position; (c) a first jaw member coupled to the actuating member and housed by the clamp housing, the first jaw member including (i) first and second bedrail portions and (ii) a first jaw member guide element configured to interface physically with the clamp housing's first housing guide element as the actuating member moves in a closing direction from the clamp-open position to the clamp-closed position to reposition the first jaw member from a first-jaw-open position in which the first jaw member does not obstruct the bedrail from entering or exiting the clamp housing's throat area to a first jaw-partially-closed position and then, via only translating the first jaw member in the closing direction, to a first-jaw-closed position in which, when the bedrail is in the clamp housing's throat area, the clamp housing's first bedrail surface engages a generally vertical outer surface of the bedrail, the first jaw member's first bedrail portion is positioned proximate to a generally horizontal upper or lower surface of the bedrail, and the first jaw member's second bedrail portion engages a generally vertical inner surface of the bedrail. (a) a clamp housing that includes (i) a throat area having a first bedrail surface and being configured to receive the bedrail and (ii) a first housing guide element; (b) an actuating member coupled to the clamp housing and configured to move relative to the clamp housing between a clamp-open position and a clamp-closed position, wherein, when the bedrail is in the clamp housing's throat area, the actuating member is nearer to the bedrail in the clamp-open position than in the clamp-closed position; (c) a first jaw member coupled to the actuating member and housed by the clamp housing, the first jaw member including (i) first and second bedrail portions and (ii) a first jaw member guide element configured to interface physically with the clamp housing's first housing guide element as the actuating member moves in a closing direction from the clamp-open position to the clamp-closed position to reposition the first jaw member from a first-jaw-open position in which the first jaw member does not obstruct the bedrail from entering or exiting the clamp housing's throat area to a first jaw-partially-closed position and then, via only translating the first jaw member in the closing direction, to a first-jaw-closed position in which, when the bedrail is in the clamp housing's throat area, the clamp housing's first bedrail surface engages a generally vertical outer surface of the bedrail, the first jaw member's first bedrail portion is positioned proximate to a generally horizontal upper or lower surface of the bedrail, and the first jaw member's second bedrail portion engages a generally vertical inner surface of the bedrail.