Part of what makes your question a bit tricky to answer is that hidden beneath it is a much more general question--"How do I do phonetics?"--which is obviously an absurdly broad question. The issue is narrowed by the fact that you are focusing on a concrete problem, but addressing it still entails a lot of knowledge about how speech and language work.
That said, several valid points have been brought up in comments and in @Nick Anderegg's answer, and I thought I'd take a crack at formulating a cohesive answer that incorporates the various points and hopefully clarifies some issues for you. But after reading my answer you might find that you have a bunch of follow-up questions!
Remember that the phonemes /e/ and /ɛ/ are phonological constructs. As @TKR notes, they are only meaningful inasmuch as they contrast in a particular phonemic inventory. The symbols a phonologist chooses to use for the phonemes are really just mnemonic labels--the symbol e indicates that the vowel in question tends to be realized as higher (and maybe fronter) than the one labeled as ɛ given the same environment. Just to be clear: in a given language, /ɛ/ may have an allophone that is just as high/front as or higher/fronter than one of the allophones of /e/. In such a case, the phonologist might characterize the allophone as [e], i.e., she might say that /ɛ/ is realized as [e] in certain environments.
So hopefully at this point you understand how the names of phonemes are purely mnemonic. But even the labels assigned to phones, such as [e] and [ɛ], do not have an absolute meaning in a vacuum. They are still labels of relativity imposed on the realizations of vowels by the phonetician, and they are language-specific. The phonetician measures the formants (and other parameters, such as duration) of many instances of the vowels in question, and she sees that they appear to cluster into two groups when everything else about the environment is the same (including, for example, the surrounding phones). The ones in the higher/fronter group get labeled as [e], while those in the not-as-high/not-as-front group (that are still higher and fronter than other vowels she's measured) get labeled as [ɛ].
Remember that IPA is language-dependent, a point I harped on in my answer to 'How can I differentiate between syllable-initial ɣ and ə using Praat or other software?'. Even if Language A has vowels that are conventionally labeled as [e] and [ɛ], respectively, and Language B also has vowels labeled as such, it doesn't mean that [e] in Language A will be the same as [e] in Language B, even in the analogous environment. Icelandic is usually described as having [e] and [ɛ], as is my dialect of English. But when my Icelandic friends produce their version of the English word bed, I hear it as bad, because the formant values and relative durations of their version of [e] in that word are more appropriate for the [æ] (the allphone I have for /æ/ between /b/ and /d/) in my dialect. Note that Icelandic doesn't have a phoneme that corresponds to my /æ/, i.e. they have more height leeway when it comes to their front-ish vowels.
Are you starting to see why it doesn't make so much sense to talk in absolute terms of whether you "have [e] or [ɛ]" in your dialect? There aren't "cardinal" measures of height/backness/duration for [e] and [ɛ]--only language-specific relative tendencies. What you can do is try to see how your pronunciations of certain vowels differ from those of native speakers in the target language.
On the perceptual front, you could do the following:
- Find a bunch of minimal pairs that alternate [e] and [ɛ] in the target language.
- Pronounce them with your one front mid vowel and ask native speakers which word of the relevant minimal pair they perceive you as saying. You'll want to ask a number of native speakers, as their intuitions might vary.
- Analyze their responses. If most of the responses tend in one direction over the other, it will be a hint that your one mid front vowel tends to map to that vowel in the target language. If the responses are more or less 50-50, it indicates that your vowel tends to hover somewhere in the middle of where they would expect their native [e] and [ɛ] to fall.
It's not as straightforward to try to approach the issue from an acoustical perspective; @jlawler is right when he points out that there are a vast number of factors that influence phonetic measurements like formant values and relative duration. Don't fall into the trap of thinking that you can measure a bunch of native speakers' formant values and compare theirs to yours! If Speaker A and Speaker B are native speakers of the same dialect of the same language, but they have different head and throat shapes, the formant values for [e] produced by Speaker A will be different from the formant values for [e] produced by Speaker B. In fact, the formant values for Speaker A's [e] could map onto the formant values for Speaker B's [ɛ], even if both speakers agree that A is producing [e] and B is producing [ɛ].
Here's what you can do on the acoustic front:
- Record a bunch of repetitions of a bunch of [e]/[ɛ] minimal pairs for a bunch of native speakers. Record them spoken in a bunch of environments (surrounded by different phones, in isolated words, in words in frame sentences, in careful speech, in faster speech, etc.).
- Measure the formant values and relative durations (e.g., the ratio of the vowel duration to the duration of a control segment somewhere else in the utterance) of the target vowels.
- Analyze the results and look for patterns.
- Record yourself trying to produce the minimal pairs, and practice differentiating them such that you eventually find the same relative patterns in your own formant values and relative durations.
I hope most of that makes sense. As this answer is getting a bit unwieldy, I'll leave it to you to post follow-up questions you may have about the best way to measure formant values in vowels or how certain parameters like F1 and F2 change according to changes in tongue height/backness!