To stabilize a trait when it is only showing in one cultivar, I would self it. And then make sibling crosses and back crosses with the best of your results. You can also make the same cross that produced the trait to see if you can produce the trait again and hopefully have two different plants with the same trait. Thankfully, when new traits arise from hidden genotypes, they are likely to occur in several gardens at once. So if you keep in touch with hybridizers in some way, you will hear of other plants you can use.
These are excellent strategies.
They work on the genetic differences that are present in the cultivar that shows the characteristic and its relatives. As an example lets say that the characteristic is affected by three genes in the plant showing it. To make things easier lets assume it is a diploid and that the three genes are labelled 1, 2, 3. The plant that shows the characteristic is genetically 1 +/+, 2+/-, 3+/- (where plus strengthens the appearance of the characteristic and minus does nothing to the characteristic and four or more pluses cause the characteristic to be visible to some extent). Selfing it, making sibling crosses and backcrosses have as their goal to produce plants that are 1 +/+, 2 +/+, 3 +/+.
There is another strategy. In this case the plant with the characteristic is crossed to as many unrelated plants as possible. The goal in this strategy is to find plants that are genetically different but can add to the characteristic. For example. there might be a plant that is 1 -/-, 2 -/-. 3 -/-, but 4 +/- and 5 -/-. There might be another plant that is 1 -/-, 2-/-, 3 -/-, 4 -/- but 5 +/-. Neither of these two plants will show any sign of the characteristic but some of the seedlings from the cross of the plant showing the characteristic with the first plant will be 1 +/-, 2 +/-, 3 +/-, 4 +/-, 5 -/- and will show some of the characteristic and some of the seedlings from the similar cross with the second plant will be 1 +/-, 2 +/-, 3 +/-, 4 -/-, 5 +/- and will also show some of the characteristic.
The original plant showing the characteristic was genetically 1 +/+, 2+/-, 3+/- , 4 -/- 5 -/-. Crossing the seedling, 1 +/-, 2 +/-, 3 +/-, 4 +/-, 5 -/- with the seedling 1 +/-, 2 +/-, 3 +/-, 4 -/-, 5 +/- will produce some offspring that are 1 +/+, 2 +/+, 3 +/+, 4 +/-, 5 +/-. Working with seedlings with those genes would then produce seedlings that are 1 +/+, 2 +/+, 3 +/+, 4 +/+, 5 +/+. Seedlings with this genotype could then be used in crosses to test yet other unrelated cultivars looking for any that might carry 6 +/- or 7 +/-, (and not ever show the characteristic) etc. to bring into the line. The intention in this strategy is to find other genes that affect the characteristic and accumulate them in the line. As more genes are accumulated the characteristic becomes more stable or more extreme, etc.
