I personally have ~1 years worth of cash savings. I have it as an emergency fund plus slush fund for any big purchases so I won't have to sell my stocks. For instance, I was able to purchase a new HVAC for my house without having to move any money around or sell any investments. I've just been slowly replenishing my savings alongside investing since that purchase.
If I was you, I would definitely make sure I have at least 6 months savings and then maybe split investments/continued savings until I was at a point I was comfortable with in terms of savings (whatever it is that you decide that value is). Also, you say your savings account pays low interest. I would look to see if there are any High Yield Savings Accounts available in your country. Here in the US you can find accounts paying ~5% which is great for an efund.