Ask HN: How to Transition from Buildings Engineering to Software Development?

I've been working for a large engineering consulting company for the past 4 years as a graduate and I'm looking to jump to software engineering or similar.

I have degrees in Aerospace Engineering and Mathematical and Computer Science, majoring in statistics and applied maths. I'm proficient in Python and Java, I've worked on a number of smaller side projects.

The biggest obstacle for me is finding an entry level job that accepts non-recent graduates. I've been knocked back by a number of companies for not being a recent (<2yr) graduate.

Does HN have any advice?