There are potentially options that could work for you, including:

1) Just charge from an existing 120V outlet. If you average under 50 miles of driving per day this might work just fine.

2) Reshuffle existing circuits to get even a 20A 240V circuit for charging. This provides more than double the charging you can get from a 15A 120V outlet and may be possible by combining some existing circuits with tandem breakers. A lot of electricians will just quote you on going straight to a 50A or 60A circuit, but most chargers are configurable to lower levels and it can save you a lot of money to stick to a smaller circuit.

3) Share a circuit with a dryer using a device that automatically switches power off to the car while the dryer is running, there are a few brands of these.

4) Get an EV charger that supports load management, using a current sensor on your main breaker. These can dynamically adjust how much current the car draws to avoid exceeding the max current of your panel.

5) Electric meter socket adapter, these connect between your utility's electric meter and its socket and provides an EV charging port that doesn't go through your household panel at all.

Can't you charge it from a wall outlet but slower (tier 1 charging)? If you have a dryer in your garage you can get a charger that plugs into that, taking turns with your dryer.

Technology Connections did a video essay on this and concluded that for most people, you get plenty of range for everyday use just by leaving it plugged into a normal socket overnight, and just occasionally using a supercharger when you're doing long trips. Of course if your commute is particularly long maybe this doesn't apply to you.

Range anxiety is definitely a real concern about switching to EV but apparently most people don't find it that bad once they actually get one. Besides, there won't be any gas soon anyway.

You don't need an EV charger. I charge my car from a normal wall socket over night, limited to 12A. That gives me 60 miles of range over night.

Depending on how much you drive per week, a 120V charger might be viable. My new house has the same problem (I'd have to upgrade the electric panel for my car due to its age), but I've been using the normal 120V charger to charge it, and haven't yet run into a case where I couldn't charge it enough.

Assuming you have a 200A service, you don’t need to replace your panel. You can remove two single pole breakers or one two pole breaker, install a 100A two pole breaker in the empty space, buy some 3/#1 copper SER cable (two hots and one neutral for 240/120V) and use it to feed a $125 100A load center without a main breaker.

You need to use #1 copper for a 100A subpanel, you can use 3/#2 aluminum cable for a 100A service entrance but there’s an exception that allows using wire rated at 83% of its ampacity for a service entrance conductor which does not apply for subpanels.

Refeed the two removed breakers out of the load center and add a 60A two pole breaker to feed the EV charger, if it’s a cord and plug charger that you plug into a NEMA 14-50R you’ll need a GFCI breaker, if it’s hardwired you don’t need a GFCI breaker.

The material can be purchased for less than $500 for this project, if you have to hire out the labor it’ll be considerably more expensive, I’d estimate one day for an electrician or around $1200-1800 depending on where you live.