The first thing to note is that the words major, paper, hate, mate have what is called a 'diphthong' in standard varieties of English. A vowel can be either a monopthong or a diphthong. A monopthong is a vowel sound in which the speech organs (hereafter articulators) remain more or less in the same position throughout the articulation of the vowel. That is, the vowel quality remains constant.
A 'diphthong', however, involves a change in the position of the tongue, thereby changing the vowel quality. The articulators glide from one vowel position to another.
Paper, major, mate have the diphthong /eɪ/ whereas pepper, measure, met have the monopthong /ɛ/ (sometimes transcribed /e/ for convenience). The [e] in some of the words you mentioned is a part of the diphthong /eɪ/.
Also note that /phonemes/ can be [realised in more than one way].
So when pronouncing measure, bed, pepper etc., make sure your tongue remains in the same position throughout the vowel sound. In some American accents, major, mate, paper have the long monopthong [e]. The main difference between [ɛ] and [e] is vowel openness; [ɛ] is more open than [e].
If you understand IPA, here are the transcriptions of 'day' (/eɪ/) in different American accents:
- Standard US: [dɛ̝ˑɪ]
- Ohio: [de̞ˑɪ]
- Chicago: [de̞ˑɪ]
- Boston: [deˑɪ]
- New York: [de̞ˑi]
- Alabama: [dɛ̝ˑɪ]
[All these transcriptions are from Sound Comparisons]