The most common and most likely answer to this crossword clue is the 8 letter word ESCALATE
I've seen this crossword clue in Independent
E
S
C
A
L
A
T
E
More clues leading to the result ESCALATE
- Key on a dead mushroom
- Living in English castle somehow is a step up
- Grow roughly fifty for the first time in grounds
- Step up key piece of activity after usual close of business?
- Discontentedly expostulates about dead mushroom
- Increase key activities, initially in the evening
- Intensify destruction of a celesta
- Rocket, key one, behind time
- Note a series of musical notes contains a time increase
- Alec’s tea involved mushroom and rocket
Buddy explains!

The straight part
We know that "Step up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case step up = escalate.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does ESCALATE mean?
ESCALATE verb- To increase (something) in extent or intensity; to intensify or step up.
- In technical support, to transfer a customer, a problem, etc. to the next higher level of authority