All marriages should immediately be subjected to the annulment process before the wedding bedding to determine if there is actually a valid marriage.
To be sure it's a bit inconvenient to put off the wedding trip and so forth. But since lots of dioceses now make a year wait mandatory, just have the wedding at say 9 months and spend the next three month determining if the marriage is actually valid.
The current annulment process is supposed to presume the marriage bond does exist, but it's obvious the process is currently disposed against that presumption. So as a means of countering this current error in the disposition, if the process of determining if a marriage is valid is immediately after the marriage vows are taken then the process would in turn be as a result disposed toward defending the bond because all parties involved would want to prove the bond does exist.
If someone came back later and tried to get an annulment citing information that wasn't given during the first process, it would not be looked at kindly because the tribunal has a stake in holding up its prior ruling. Institutions always defend their own against outsiders and the same could be expected of the tribunals.
As it stands, the disposition is against the marriage, and so that is where the change needs to be made, and the best place to make it is where there would be the least resistance. The dioceses already go way overboard in their marriage preparation only to look bad afterward because of all the annulments, so this would be a way for them to save face.
People act for the end they desire. Which in turn means that all the annulments petitioned for reflects a common desire to abandon marriages they are currently in, no different than the reason divorce is so common.
The annulment process is currently sought because someone wants out of the marriage. Not unlike those who conversely want to stay in a marriage which they know is not valid in turn seek to have their marriage blessed so that their marriage is valid. For instance, know a couple that was first married by a judge, then by a schismatic church and finally by a FSSP priest. They didn't seek an annulment, because what they wanted was to be married.
So what needs to be done is to shift the focus away from abandonment to the opposite using the same process that is currently working against marriages.
What do I think about priests who give scandalous advices? St. Matthew Chapter 18
My other posts on the annulment scandal.
The annulment scandal
"The floor of hell is paved with the skulls of bishops"
My other solution to the annulment scandal
Never trust a priest with your marriage problems
Stressing the unitive aspect of marriage only makes the problem worse
A solution to the annulment scandal
Marry a hot Catholic babe with a 2-year long marriage contract,
and receive a guaranteed annulment because of "lack of due discretion" when the contract runs out