After learning a set of the alphabetical letters in sign language, they then would fingerspell out a series of the alphabeticall letters of a word or name. And, they typically would try to pay attention to each of the letters spelled out by a signer.
But, in reality, native and fluent signers fingerspell naturally. Actually, they are not producing nor reading letter by letter. They look at the whole configuration of a fingerspelled word, just the same way a hearing person listens to the whole word, not a phoneme by phoneme (or sound unit).
Now try fingerspelling the word "fly". You might consciously spell it F-L-Y -- these very same handshapes as you'd see in a manual alphabet chart. But, if you're capturing a fingerspelled word in daily real-life settings, you'd see such things as this image.
This signer (myself) was completely unaware and natural as I fingerspelled it in front of the camera. The result shows that there is no letter "L" in it. It only shows F-ILY-Y rather than F-L-Y! But, to the eye of a native signer, it looks perfectly right.
What's going on? Does the sound unit "O" sound the same in "pot" and "foul"? No? Exactly the similar idea in sign language. Each unit of a word is affected by the other units before and after. The unit is stretched, pulled, rotated, twisted, and such by the previous and the next units of a word, whether signed or spoken. It's known as co-articulation in linguistics.
Want some more examples?
Now try fingerspelling "JOHN" letter by letter as you've seen in a fingerspelling chart. In real life, observe an ASL speaker articulating "JOHN" below.
You won't see the same configuration of some letters as you see in the chart of individual alphabtical letters. The palm orientation of the letters O and H are different. See the video again in slow motion below.
These are only a few examples. When capturing fingerspelled words into still images, you would find lots of different shapes (including orientation) from the original print. It's something similar you'd find in spoken words.
Fingerspelling aside, one example I can think of in signing ASL is the co-articulation for TELEPHONE BOOK. The palm orientation is slightly changed when signing TELEPHONE (more horizontal orientation) BOOK (orientation influenced by BOOK).
Armstrong, David; Stokoe, William; Wilcox, Sherman (1995). Gesture and the Nature of Language. Cambridge University Press. Pp 106-114.
Also see phonology: the study of the smallest units of language.