I have a basement situation where the waste pipe is above the washing machine. So the water from the washing machine is drained into the slop sink then drained into a pump to push the water to the waste pipe. Can you recommend a pump that will handle the water directly from the washing machine and then to the waste pipe? (Something small enough to fit in a cabinet under the sink.)

A washer puts out a very high flow rate and a small under the cabinet pump might not be able to handle it. How would you connect it? You need an air gap between the washer pipe and the drain system, otherwise you can backflow from the sanitary sewer. Most washer pumps will produce a pretty good head pressure. What is the lift you need and why don't you want to keep what you have?