Has anything changed in the garden - any new animals you have seen taking up residence, any new roaming cats, any new noises?
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A small number of spayed females develop incontinence later in life. Many who do can be managed with meds.
Now, I know you say that Lizzie is not incontinent - and thank goodness for that!
However, the urge to urinate more frequently, even while still being able to exercise bladder control, can be a sign of a developing condition.
That she urinates - sometimes twice - right away on being let out might (and only might) speak to a developing condition. Of any of the causes.
OR
It might simply be age related, either the beginning of CCD or simply 'senior moments', where our oldies start doing odd stuff but the oddity stops short of CCD.
OR
It could even be a combination of all of the above.
Were it me, I'd probably:
Treat it as if there were some sort of age related cognition thingy going on, and work on relaxation and stress reduction at night.
If the behavior persists, see the vet again. Both to review any other possible medical explanations for increased asking to go out, and to talk about CCD.
And obviously, if any other medical related symptoms crop up, see the vet sooner rather than later.
Oldies - gotta luv 'em.