That's not all. Some Russian consonants, namely "ш", "ж", "ц" don't have their soft pairs. So, the letter "е" after them doesn't soften these consonants:
Жесть - [жэст']
Цель - [цэл']
Шест - [шэст]
Actually, the consonant sound [ш] has a soft pair, but this sound has its own letter "щ" so the letter "e" after "ш" doesn't soften this consonant.
There are also some words in Russian which are borrowed from foreign languages that have latter "e" in place of original Latin letter "e", though in such words it's often doesn't soften the preceding consonant. For example:
Тест is pronounced [тэст]
Прогресс - [прогрэс]
Партер - [партэр]
Although the last rule isn't very strict and there can be variations. For example, some people could pronounce the word "прогресс" as [прогр'эс] (with the soft "р"), some pronounce it as [прогрэс] - with the hard "р". The same story happens with such word as "энергия".