Please fill out the fields below so we can help you better. Note: you must provide your domain name to get help. Domain names for issued certificates are all made public in Certificate Transparency logs (e.g. crt.sh | example.com), so withholding your domain name here does not increase secrecy, but only makes it harder for us to provide help.
Let's Encrypt certificates do not have to be expired before renewing them. In fact, the recommended renewal schedule according to Let's Encrypt is 30 days before expiration (60 days after issuance). Overlapping certificates can both be valid at the same time, so you don't have to have an outage or downtime. The new certificate doesn't invalidate the old one just by its existence.
Beyond that, it would be helpful to know which Let's Encrypt client application you're using, since there are about 100 different options and they all behave differently (including about 5 or 6 different applications that are IIS-specific).