Ok, let's start. See if you know a member of the private firm. If you do, call he/she and ask about the position that they had open. Point out that you are looking to increase/enhance your engineering background and felt that your background and this position would be a good match. MAKE NO MENTION that you are dissatisfied with your current position-- doing that will suggest that you will be unsatisfied with their position also. Always positively point out that your current position is a great job, however, you wish to broaden/specialize/focus (what ever it is) and this new position suggests that you will do that.
Identify that your public sector experiernce is valuable and then tell them HOW and WHY it is valuable-- to the firm. The question that you are answering here is, "Why should we hire you?" Well, you have been in the public sector for xx years, know xx about yy and these factors are important to the firm as they continue to focus on zz types of work.
The answer may be that the position is filled, thanks for the call. Be gracious, tell them thanks for taking the time and hang up. Whomever you visited with, get his/her name, then send a follow up thank you note for taking their time to visit with you. This does not need to be anything fancy or elaborate, however, sincerity is the key here. Enclose a resume with it, with something on the note like this, ........." I have attached a resume for future reference. If the opportunity of another position opens up, I would appreciate the consideration of being reviewed for such a position."
If you are in the public sector, go to a PE meeting and poke around and ask others if there are any other opportunities in their firm. Don't be bashful here, Start out asking about how business is, then just ask, "are you currently searching to add additional professional staff?" If the answer is yes, then puruse it, pointing out, succinctly, your background and asking whom you may wish to visit with about further exploration of the position. IF no, again, say thanks and keep talking.
If you are not a member of the PE club in your community, might be a good idea to join. Get a membership directory and then just call the ranking official at the firm and ask if ther are "any opportunities" for someone with your background. You need to cogently and succinctly identify what that background it and then listen.
Sooner or later, this approach will yield fruit-- make take a while, however, the approach works!! Give it a try and report back on your success.
