Starting reta is different to swapping from tirz, starting at 4mg is probably a bad idea , swapping is much less likely to be a problem. The last time I suggested just swap over at 6-8mg , I got several critical replies , some had pretty good science behind them, and on average reta has more side effects than tirz.
It also depends a lot on what side effects you get now from 15mg of tirz and what it was like as you increased doses, the more side effects you got from tirz, the slower and more careful you should be. In general the slower and more careful you are, the less likely it is you will get sudden severe side effects, probably does not make much difference to overall amount of side effects for a given dose though.
So you can just gradually introduce reta while dropping tirz or just stop and swap, I do not think it is a big issue.
Also depends on how much of a problem it is to be on a lower dose of reta for a while, while swapping over and if regaining some weight in that time is a major issue.
I would not start higher than 4mg on a straight swap, in general if you want to escalate doses faster than every 4 weeks or so, the lowest risk approach is smaller doses more often, so every 2 or 3 days or twice a week, so each small extra dose will likely give you some side effects or not and help decide if continuing to increase doses is going to make them worse, and usually if you overshoot your tolerance it should take only a few days for blood levels to drop back to pre last dose levels. So something like 2mg 2 x a week, and increase 0.5mg per dose every week or 2 , assuming there are no side effects. If it is starting to kick in and have effects like appetite suppression or side effects like nausea, then you need to slow the increases right down. Something like that would get you to 6-8mg in 2 to 4 weeks, with fairly low risk of unexpected severe side effects. Check out glp plotter so you can see what different doses do to blood levels over time, you need to have some understanding of what is going on if you are going to deviate from the standard schedule. Also do not do this if you have not previously been on a glp drug. Once you are at a dose you are happy with and want to swap to weekly, just gradually increase one dose and decrease the other one.
When I swapped from ozempic to tirz I just added an extra mg every day or two for a while, until it seemed to be suppressing appetite a bit or causing mild nausea, but it got me to 15mg a week in about a month with no real problems.