Raspberry plants have been termed a bramble for their voracious growth habits. You might have one plant, but if you leave it alone in the right conditions, it will produce three to six new daughter plants in one growing season. Leave it alone again for a season, and you might have 36 rooted plants growing in that one spot. Pruning can eliminate most of this growth, forcing the plant's energy into producing more fruit, but if you want to increase your raspberry patch fast, just transplant all these daughter plants or suckers to a new location.
Clean up your raspberry patch in the fall before the freezing weather comes, if you haven't done so already. Some gardeners will prune back their canes after the fruiting season, while others will allow the canes to grow and store up nutrients for the next season. Cut out all the previous year's canes, as they will not produce any more fruit.
Lift any suckers that have grown off the arching canes by holding the stem with your gloved hand and lifting them as you dig under them with a hand shovel in the other hand. There should be a good sized root mass if it had a couple of months to grow. Cut the cane so it is removed from the mother plant.
Dig a new hole about 12 inches from other plants and about 6 inches deep. Place the new sucker plant into it. Replace the soil around the roots. Fill in the hole and tamp it down with the heel of your shoe. Trim back the cane to about 6 inches. You might get some fruit from it the next growing season, but the best crop will be in two years.
Treat the daughter plant now as a raspberry bush, pruning it back and staking or trellising it as any other of your raspberry bushes, depending on the variety you are growing. Keep it watered and mulch it with a few inches of leaf mold to keep the pH to around 5.5 to 6.5.