Is there such a thing as a multimeter with needle prongs as small as an exacto knife tip?
I have replaced the gross sized meter probe times on some of my meters with short sewing needles. These have the advantage that they are very sharp and very hard steel so they stay sharp a long time. 

In my case I found the needles too small diameter for the collet on the end of the probe lead so I tightly wrapped 30awg bare wire around the end of the needle opposite the point. The wrap length was about 0.5 inch. Then with good flux and a hot iron I soldered the wrapped wire in place to increase the overall diameter of the needle body. 

One set of probes I prepared like this about 8 years ago still has nice sharp ends.