The short, sort of accurate, answer is: all unstressed syllables in English have the "schwa" . The exceptions are that final unstressed syllables can sometimes have full vowels (e.g., potato) and [i] can often be unstressed even in the middle of words (e.g., radiate).
The longer, more accurate answer relies on the distinction between narrow and broad transcription.
Unstressed vowels in English are quite variable. The same speaker will pronounce the vowel  in the second syllable of enough much the same way every time, but the schwa in the first syllable can be pronounced very differently on different occasions, sometimes even resembling full vowels like , , or . [Note]
One common solution to this problem is to transcribe the "er" sound with the special IPA symbol . Unfortunately, there are no special symbols to solve the similar problem with [n], [l], and [m] -- for example, in normal speech there is simply no vowel between the [t] and the [n] of button, despite the usual broad transcription . We will return to this problem later in the course when we discuss "syllabicity". For now, it's easiest to continue using schwas in broad transcriptions of words like these.
The classical minimal pair to illustrate this distinction,
in dialects that make it, is roses versus
Rosa's. You could record a speaker of such a
dialect saying roses and Rosa's a hundred times
each and plot on a graph the position of the speaker's tongue during the
final vowel. There would be a large cloud of different positions
for roses and a large cloud of positions for Rosa's
-- there would be a large area where the two clouds overlapped,
but it would still be clear the clouds had different centres.
If you did the same graph for a speaker of a dialect that
doesn't make this distinction (like me), the two clouds would
overlap so much that there would be no justification for saying
that the two words had different vowels.
In transcribing roses and Rosa's, the difference between narrow and broad transcription is again relevant.
The transcriptions in the textbook fall somewhere in between. Rogers generally uses schwa for the vowels of unstressed syllables, but occasionally uses  in words where dialects which make the schwa/barred-i constrast would use barred-i, e.g., relax. (This is not entirely consistent. Even if a speaker does have two clouds for their unstressed vowels, the grounds for identifying the higher cloud with the vowel of hit are no stronger than the grounds for identifying the lower cloud with the vowel of cup.)
In my transcriptions, I will only use  for
neutral unstressed vowels, i.e., for all unstressed vowels that
are not full vowels, like the [i] in happy or [o]
in potato. On assignments and tests, using schwa
in broad transcriptions will always be acceptable. It's also a
good habit to get into, as one way of unlearning habits that
might carry over from English spelling.