Go everywhere with glass firstly (or bottle when drunk) (5-4)

I believe the answer is:

globe-trot

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'go everywhere with' is the definition.
I can't tell whether this definition defines the answer.

'glass firstly or bottle when drunk' is the wordplay.
'firstly' indicates taking the first letters.
'when drunk' indicates anagramming the letters (a drunken or disorderly form of the letters).
The initial letter of 'glass' is 'g'.
'or'+'bottle'='orbottle'
'orbottle' with letters rearranged gives 'lobetrot'.
'g'+'lobetrot'='GLOBE-TROT'

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I've seen this clue in The Telegraph.

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