Linear B is an interesting example for this:
This system was apparently designed for a non-Greek language, as it did not fit the sounds of Greek very well. In fact, it is likely that Linear A was used to write the pre-Greek language of Crete, and the incoming Greeks adopted this writing system for their own use, but without changing how the system fundamentally works. In doing so, they developed "spelling conventions" to represent sound patterns found in Greek but not in the syllabary.
First, there are many Greek sounds that are missing in Linear B signs, such as [g], [kh], [gw], [b], [ph], [th], and [l]. To solve this problem, signs for similar sounds are used instead: p-signs are used for [p], [b], and [ph]; k-signs are used for [k], [g], and [kh]; t-signs are used for [t] and [th]; q-signs are used for [kw] and [gw]; and r-signs are used for [r] and [l]. However, while this convention was likely easily understood by ancient Mycenaean scribes, it took modern scholars a lot of theoretical analysis and work, plus comparison with later Greek dialects and reconstructed Mycenaean words to rediscover how this system works
And more relevantly:
Another inadequacy stems from the fact that Linear B signs usually represent Consonant-Vowel (CV) syllables, but the syllabic structure of Greek allows initial consonant clusters, ending consonants, and dipthongs. In the case of a syllable with a initial consonant cluster, individual consonants in the cluster are written by a CV sign whose vowel matches the vowel of the syllable. Therefore, for example, the word tri is written as ti-ri, and khrusos as ku-ru-so. In the case of ending consonant, the situation becomes more complicated. Ending consonants such as [l], [m], [n], [r], and [s] are not usually written, whereas other consonants such as [k] and [p] are written in a way similar to initial consonants.
So one way to represent consonant clusters in a syllabary is to pick a "dummy vowel" that gets read over when the word is pronounced, as in Linear B, or Japanese katakana when writing foreign words.
The question doesn't mention just how complex the syllable structure is; if it's not too complex, you can also do what Japanese hiragana does with the ん character and have a single character that represents a syllable-ending consonant. Or something like what Brahmi does for vowels and nasals, adding strokes and dots to represent them. Maybe if kra, tra, and pra are common syllables in your language, you could have a stroke that represents an r between the consonant and vowel, and add it to the characters for ka, ta, and pa.
