Stretching causes both acute (short-term) and chronic (long-term) increases in flexibility. You get a very big, but very temporary increase in flexibility, as well as a much smaller but more permanent increase in flexibility. So you will always be more flexible right after you've stretched than the next day.
You're not doing anything wrong, and if you want to make the long-term increases in flexibility come quicker, all you can do is stretch more often. If you will need to demonstrate flexibility at a specific time (e.g. for sports performance), then just make sure you stretch that day before you need to demonstrate maximum flexibility.
Here are some examples of evidence for this effect:
- Lew Hardy (1985) Improving Active Range of Hip Flexion
In these results, the "Day 6" measurement was taken immediately after the final session of a 6 day stretching program, and the "Day 7" measurement was taken a day later. You can see how the measurements taken immediately after the final stretching session are much greater than those from 1 day post-training. The study explains that this is the difference between the acute (long-term) and chronic (short-term) affects of stretching, stating: "The Day 7 scores were therefore regarded as a measure of the long term, or retained, gains resulting from the different treatments. However, on Day 6 the experimental subjects were posttested after stretching. The Day 6 test for the control group was administered in the same way as on Days 1 and 7. The Day 6 scores were therefore regarded as a measure of the immediate (warming-up) benefit of the different treatment methods."
- Bruce R. Etnyre & Eva J. Lee (1988) Chronic and Acute Flexibility of Men and Women
Using Three Different Stretching Techniques
In this study, measurements were also taken before ("Pre 12") and after ("Post 12") a final training session. It isn't clear from the study text whether the week 3, 6 and 9 measurements were taken before or after stretching, but from the results it appears that they were taken after stretching (as otherwise they would indicate a loss of chronic flexibility between weeks 9 and 12). Once again we see a large jump in flexibility when measured immediately after training, but when flexibility is measured when the trainee has not recently stretched, they are seen to have smaller increases in flexibility. (But they are still more flexible than they were before the beginning of the program.)
For more accessible further reading on this topic, I'd recommend Stretching Scientifically, by Thomas Kurz. On the topic of short vs long term changes in flexibility, he writes: "If you need to perform movements requiring considerable flexibility with no warm-up, you ought to make the early morning stretch a part of your daily routine (Ozolin 1971; Wazny 1981b). [...] The purpose of this stretching is to reset the nervous regulation of the length of your muscles for the rest of the day." (Note that Kurz is Polish and developed his career during the Soviet Union, hence most of the papers he cites are written in Polish or Russian.)