How to manage imposter syndrome during the job hunt?
Actually user Bob Brown has the germ of the answer in his comment. Having imposter syndrome doesn't mean that you are an imposter. It is a mental condition, nothing more. There are many such conditions that can serve as handicaps at your stage of career. Extreme introversion is another. Even, oddly, is breaking your leg so that travel becomes impossible for a while.
How do you deal with these things? The answer is "Every day, little by little".
Imposter syndrome affects how you feel, but it doesn't have any necessary effect on what you do. You can become the success you'd like to be if you just act as if you don't have these feelings. "Fake it till you make it" is actually good advice for such things.
You need to send out letters and get letters of recommendation and make your case for your success in a job whether you feel confident about it or not. So, just do that. If you also feel like it is just a waste of time because you aren't worthy then it will only become a self-fulfilling prophesy if you let the feeling affect your actions.
But if all you need, right now, is to feel better about yourself, then I'd suggest going back to that advisor and asking for her/his most honest assessment of your skills/position/prospects/whatever. Just a little dopamine hit to keep you smiling while you write those dam' letters.
My issue, actually, was the extreme introversion thing. It held me back until I got some advice that said "ignore it". Act like an extrovert. It was hard and took a long time, but few people other than my spouse think of me as introverted anymore.
In your case, just ask yourself, how would I act if I weren't an imposter? Then do that.
(every day, little by little)
You need to pump yourself up. Concentrate on your positives and don't think about your negatives. If you are more confident, you interview better. It won't change something radically (if you are not competitive), but it will help you on the margin. Plus it feels better.
The job market is a more objective and larger data assessment of you. So it is natural that you may be concerned, especially versus a comforting advisor. But look at it as a challenge. If you go at it head on, you do better. Nothing to lose anyhow.
[Good luck.]