I put my legal name on the actual forms that I filled out once I got the job, and they have a copy of my passport with my birthname on it, but my preferred name went on the cover letter/ my CV when i applied, and i put in a note about my name change and pronouns. This did mean basically outing myself to my managers but I don't pass very well so it was going to happen anyway.
It also depends on how changing your name works where you live -- i have a deedpoll which is free to make and legally binding in the UK (in fact I didn't even really need to make one of those because i live in scotland and we have our own laws, but most places ask for one anyway) so i could give my work a copy of that if they're concerned. I also put M on the forms and I have no GRC, but no one has asked any questions, but that may be an oversight on their part.
Also, I was pretty confident that the place i work wouldn't turn me down simply for being transgender. I've heard of other people not outing themselves until after their probation period ends - so if you think the place you want to work might not want to deal with you being trans, you could use just your legal name at first and then once you have the job let them know that you'd prefer a different name. That does have the downside that everyone you work with would know your old name though - because i approached my work with my new name, only the people that handled my application and induction know my birth-name.
You could check out the places you're applying to and see if they have any transgender protocols online - bigger companies might. Or, you could email them beforehand and enquire.