According to traditional Catholicism, what is the spiritual solution for the divorcee who hasn't had their marriage annulled?

Let's start with a discussion of what the situation is, as the Church traditionally has seen and taught it:

The official teaching of the Church is that civil divorce is usually immoral in itself, but may be morally tolerable under certain circumstances:

If civil divorce remains the only possible way of ensuring certain legal rights, the care of the children, or the protection of inheritance, it can be tolerated and does not constitute a moral offense.

(Catechism of the Catholic Church, paragraph 2383)

And a Catholic who is "the innocent victim of a divorce decreed by civil law" (Catechism paragraph 2386) is not committing a moral offense.

However, even in these two cases, the Church teaches that the civil divorce does not in itself end the marriage before God - indeed, that is impossible if the marriage is valid:

A marriage that is ratum et consummatum [that is, validly constituted, and subsequently consummated] can be dissolved by no human power and by no cause, except death.

(Code of Canon Law, Canon 1141)

That being the case, the question of whether the couple are "still married"—that is, whether they were validly married in the first place—must be investigated beyond determination of their divorced status. The Church will presume, in the absence of evidence to the contrary, that the couple are indeed still married, because this is how the Church works: it assumes a marriage is valid until it can demonstrate it to be invalid:

In a doubtful matter marriage enjoys the favor of the law.

(Code of Canon Law, canon 1150)

Thus, the traditional Catholic view is that a divorced couple who have not obtained a decree of nullity (an annulment) are still considered married.

This is not a problem in itself; but if the couple is remarried (civilly—they wouldn't be able to be remarried in the Church), the Church considers them to be committing adultery:

If a husband, separated from his wife, approaches another woman, he is an adulterer because he makes that woman commit adultery; and the woman who lives with him is an adulteress, because she has drawn another’s husband to herself.

(Catechism, paragraph 2384, quoting St. Basil, Moralia paragraph 73)

Adultery being a quite serious sin, the Church teaches that the civilly remarried couple are not fit to receive the Eucharist:

If the divorced are remarried civilly, they find themselves in a situation that objectively contravenes God’s law. Consequently, they cannot receive Eucharistic communion as long as this situation persists.

(Catechism paragraph 1650)

The Catechism of the Catholic Church does offer some suggestion about what to do in such a case:

Reconciliation through the sacrament of Penance can be granted only to those who have repented for having violated the sign of the covenant and of fidelity to Christ, and who are committed to living in complete continence. ... They should be encouraged to listen to the Word of God, to attend the Sacrifice of the Mass, to persevere in prayer, to contribute to works of charity and to community efforts for justice, to bring up their children in the Christian faith, to cultivate the spirit and practice of penance and thus implore, day by day, God’s grace.

(Catechism para. 1650–51; the second sentence is quoted from Familiaris Consortio)

It is not specified what "in complete continence" means; it could mean that the remarried couple is to separate, or at least that they are to live together without sexual or romantic activity of any sort—either of which would be no doubt rough on their relationship, and certainly on any children of their union. The Church nowhere seems to teach that someone must go back to their previous marriage, though perhaps that might be suggested by individuals whom the couples could consult for guidance. It appears to prefer that, since the civilly divorced couple have given an indication that they are unable to live together, they should live separately and in celibacy—either with or without the company of their civilly-remarried spouse.

This is no doubt "a hard saying" (cf John 6:60); many in such a situation understandably feel that the Church is abandoning them. Nevertheless, the Church currently teaches that this is the only way available to insist on (a) the holiness of the Eucharist and (b) the lifelong nature of marriage. Those who are searching for a different approach do so out of pastoral feelings for Catholics affected by the situation, which they are trying to reconcile with the formal teachings of the Church.