If a vertical mill table is loose and has side play, the best solution to alleviate side play is to:

Tightening the taper gib adjustment is the most effective way to alleviate side play in a vertical mill table. The taper gib is a component that helps secure the movement of the table and provides stability. When the table has side play, it typically means that the gib is not properly adjusted, allowing for excess movement.

By tightening the taper gib adjustment, you effectively reduce the clearance between the table and the ways it slides on. This adjustment brings the table into a more precise alignment, eliminating the excessive side movement that can impact machining accuracy and performance.

Other methods, such as replacing bearings, while potentially important if a bearing is damaged, do not address the specific issue of side play directly and might not be necessary if the gibs can be properly adjusted. Lubricating the table can improve smoothness of motion and help with wear but does not solve the side play issue itself. Increasing the table load may create more friction that holds the table in place but does not resolve the actual mechanical play present in the setup. Therefore, adjusting the taper gib is the targeted solution that directly addresses the problem.