Give the lawn a makeover
Dig round the stump to expose the roots and cut through each one, removing a section to make life easier. There’s usually one infuriating root in an awkward position that holds out for a while, but a degree of patient bloodymindedness will sever it in the end.
If the stump is a big one, call in a professional (there are registers of suitable contractors), who will either dig it out by hand, cut through the roots and winch it free or use the ultimate weapon, the stump grinder, which will gradually reduce the object to a vast pile of wood chippings and sawdust. This is the option to use if you wish to carry on as if the stump had never been there.
You will need to consider access to the site, the effect on surrounding surfaces of the use of machinery and disposal of the waste. The single most important operation that will improve the quality of any lawn is spiking. This should be done when the ground is soft, in autumn or early spring. On a small lawn, simply walk across the lawn with a fork, pushing the tines in to their full depth at frequent intervals. On a grander scale, there are machines for hire that will do the same job. The improvement will be dramatic. If you shovel sharp sand or compost over the spiked lawn and then work it in with a brush or a specialist tool called a lute, then so much the better.
Scarifying is another job that does wonders for the lawn, by scratching out moss and thatch, and the surface tangle of dead shoots. It should be done when the lawn is hard, in the height of summer. Again, the small lawn can be treated using a spring-tined rake, providing the added bonus of tautening the stomach muscles. Those who use a cylinder mower, such as an Allett, will be aware that there is a simple comb attachment for the roller that will do the job little and often, rather than generating a monumental volume of moss each year.