Does /ɛ/ have more than one sound?

The short answer to the question of whether the English phoneme /ɛ/ “has more than one sound” is simply yes, sometimes it can — but you aren’t supposed to notice. :)

The IPA symbol you used, ɛ, represents the open-mid (or low-mid) front unrounded vowel. It is typically heard in the English words you mentioned: bet, get, employ, friend. You can also hear it in the French word meaning queen, reine, or in the Italian or Portuguese words meaning seventh, respectively settimo and sétimo.

This is the sound which many native speakers use for what is known as the DRESS vowel under the Wells Standard Lexical Sets for English.

But the thing is, that doesn’t mean that phonemic /ɛ/ is always going to come out as exactly [ɛ] phonetically in all possible words irrespective of the surrounding phonological environment, or in the accents of all possible native speakers. It very much does not do that, although the whys and wherefores of each situation are a little different from each other every time.

Dictionaries do not attempt to represent these nuances, and this can lead people to think that everyone says that. They don’t. Every single person has their own accent peculiar to them, and many regional accents differ considerably from one another, including notably in how the DRESS vowel works out there.

You haven’t said just which speakers you were hearing with bet and get coming out closer to [æ] then to [ɛ], but this is not at all uncommon. Any number of factors can lead to it happening, but one commonly heard one occurs in the chain shift known as the Northern Cities Vowel Shift from the Inland North dialect of the United States. In a chain shift, a bunch of vowels rotate, each moving out of the way of the other. Because /æ/ moved under the well-documented phenomenon of /æ/ raising, phonemic /ɛ/ also had to move to compensate for this. Per Wikipedia:

Backing or lowering of /ɛ/

The movement of /æ/ to [ɛə], in order to avoid overlap, presumably initiates the further movement of the original /ɛ/ vowel (the “short e” in DRESS) towards either [ɐ], the near-open central vowel, or almost [æ]. As the vowel [ɐ] is pronounced with the tongue farther back and lower in the mouth than in the sound [ɛ], this change is called “lowering and/or backing”.

However, what’s happening with your second pair, employ and friend, is probably something else than what is happening with your first pair. For one thing, these will both be subject to what’s called the pin–pen merger for those speakers who have that.

For example, in Alabama the word ten can come out [tʰɪ̠n], where ɪ̠ is a retracted version of the near-close near-front unrounded vowel. And in North Carolina it can come out as [tʰɪĭən], which all decoded out means:

 tʰ     voiceless alveolar plosive              U+0074   LATIN SMALL LETTER T
        aspirated                               U+02B0   MODIFIER LETTER SMALL H
 ɪ      near-close near-front unrounded vowel   U+026A   LATIN LETTER SMALL CAPITAL I
 ĭ      close front unrounded vowel             U+0069   LATIN SMALL LETTER I
        extra-short                             U+0306   COMBINING BREVE
 ə      mid-central vowel                       U+0259   LATIN SMALL LETTER SCHWA
 n      voiced alveolar nasal                   U+006E   LATIN SMALL LETTER N

It’s not just in the American South that the DRESS vowel can shift around. In several dialects of English from places as far away from each other as Scotland and New Zealand, the word seven ends up having that “short i” vowel.

But you weren’t hearing the KIT vowel (normally ɪ) in friend instead of the DRESS vowel (normally ɛ); you said you were hearing the FACE vowel (normally e). There aren’t all that many accents where you can get a close-e instead of an open-e there, but there are some.

For example, in South Africa head is [he̞ˑd̥], which decoded is:

 h      voiceless glottal fricative             U+0068   LATIN SMALL LETTER H
 e̞ˑ     close-mid front unrounded vowel         U+0065   LATIN SMALL LETTER E
        lowered                                 U+031E   COMBINING DOWN TACK BELOW
        half-long                               U+02D1   MODIFIER LETTER HALF TRIANGULAR COLON
 d̥      voiced alveolar plosive                 U+0064   LATIN SMALL LETTER D
        voiceless                               U+0325   COMBINING RING BELOW

And in North Carolina it’s [heɪ̆əd], which decodes to:

 h      voiceless glottal fricative             U+0068   LATIN SMALL LETTER H
 e      close-mid front unrounded vowel         U+0065   LATIN SMALL LETTER E
 ɪ̆      near-close near-front unrounded vowel   U+026A   LATIN LETTER SMALL CAPITAL I
        extra-short                             U+0306   COMBINING BREVE
 ə      mid-central vowel                       U+0259   LATIN SMALL LETTER SCHWA
 d      voiced alveolar plosive                 U+0064   LATIN SMALL LETTER D

On the one hand, it’s not in theory too unusual that you’re hearing an [e] in those words. After all, it’s a sound that’s only a little ways removed from [ɛ]. You can look at [e] and [ɛ] as tense and lax versions of the same vowel, or if you prefer, as close and open versions, where [e] is tense or close and [ɛ] is lax or open.

But I think what’s happening here in these two cases is that it’s the specific kinds of changes in the preceding vowel brought about by the nasal consonant in those words which is really what’s throwing you off. Well, not throwing you off, really. Just making it sound a little different.

Both words you mention have a nasal consonant immediately following the vowel in question. Just as in Spanish, in English it is common but not universal for a nasal consonant to change the vowel it follows a little bit under regressive assimilation. It makes the vowel a bit nasalized, and a little closer and tenser. But we don’t think of that as being a different vowel, in part because English and also Spanish don’t use the oral–nasal distinction phonemically the way French and Portuguese both do.

Your name is Fabricio, which I take to be either a Spanish name or a Portuguese one, where it would be spelled Fabrício if we were being orthographically correct. This may well condition your perception of that vowel, especially if you are a native speaker of Portuguese, because it would make you hear any nasalization as a signal that it’s really a close vowel being used there.

The Spanish word tiempo [ˈt̪jẽmpo] meaning time or weather does sometimes have a nasal vowel there that’s a bit more open in some speakers yielding [ɛ̃], but they don’t think about it because it’s not phonemic. And its cognate in Portuguese, tempo, is nearly always a nasal close e with [ˈtẽpu], although in Portugal it might occasionally come out a bit more open, so more like [ˈt̪ɛ̃pu].

So my guess is that either the speakers you’re hearing say friend are nasalizing and closing their open-e a little bit, or else you are a native speaker of Portuguese or Spanish who is therefore used to hearing a nasal e as a close vowel — or perhaps both.