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So I was researching IPA a bit and tried to map the sounds of my native language to it. What I encountered was that our typical "i" is between the "i" and "y" on this chart on Wikipedia: https://en.wikipedia.org/wiki/IPA_vowel_chart_with_audio or the official one: https://www.internationalphoneticassociation.org/sites/default/files/IPA_Kiel_2015.pdf
Why is there no symbol for that sound in IPA?
This is really more a comment on Jknappen's answer, but it's too long. A premise of the IPA is that each symbol is a sound that is a contrastive phoneme is some language. The letter "i" represents a general area of the vowel space, which overlaps the space of adjacent vowel somewhat (especially "ɪ"). If you look at a range of IPA illustrations, you will note that the vowel positions vary within the vowel frame depending on language. The position of the vowel in the vowel frame is more informative than the letter used, but that spacing obviously can't be included in a transcription. Ideally, this positioning would be based on measured mean formant values, but typically they are based on subjective evaluation. Usually, the convention is to select the IPA letter that is closest to the perceived location of the given language's instance of the vowel, but you can pick a different symbol either for phonological reasons ('it acts like a front vowel') or typographical reasons ('why not just use "a"? How do you even type "ɐ"?').
In order to reduce all phonetic vowel differences to single letters, an enormous set of distinct symbols would be necessary, since there are dozens of known kinds of "i", likewise "ɪ", "e", "ɛ" and so on.
There is no such symbol because it is not needed (generally). IPA is already large enough and it has means to add more information "between the symbols".
Usually, IPA is used to write phonemes, i.e., it gives enough information on the distinctive phonetic features in a language. A lot of articulatory detail is usually omitted and left to annotations.
If you want, you can add diacritics and other marks to give a much more precise phonetic information, and IPA has means to achieve this. So you can add markers for centralisiation and other features that describe the sound more precisely.
For the highest level of precision, linguists will always resort to sound recordings or even videos.
